Reality is really an immense interconnected binary tree, and we must always travel along a single path. It has the property that reconciling paths (paths that diverged but met up again with no interaction in our universe) will again be accessible to us.
As Galileo was prosecuted for supporting Copernicus' heliocentric theory (more specifically for championing reason over faith), the Church's default stance was that the planets moved around the sky 'as if' they orbited the Sun. We are in a similar position now with many-worlds vs. 'as if' many-worlds, based on experimental evidence such as the Mach-Zehnder interferometer results.
Nothing could be further from the truth. The first proponents of the scientific method saw the process of describing the known universe as possible only because of their faith in a rational Creator, their definition of the word "faith" meaning "conviction backed by reason" (Hebrews 11). Their hypothesis was that the creation of such a rational Creator would necessarily be ordered, not chaotic as the pagans of the day believed, and that it would be possible to seek to describe the creation in terms of scientific laws and principles. By faith they understood that what is seen was not made out of what was visible. This was the basis for the birth of the scientific method.
In the days of Galileo, the Church as you refer to, was nothing more than a political militant state, opposed to the theology of the early Christians of the 1st century, and opposed to the Scriptures which exposed its hegemony. Indeed the Church would have mothers and fathers burnt at the stake for teaching children the ten commandments and the Lord's prayer. People like William Tyndale, and many other brilliant Oxford and Cambridge scholars were hounded and martyred by the Church for translating the Bible into English and circulating and discussing it in the 1500s.
While the Church may have opposed heliocentrism, Galileo defended heliocentrism, and understood correctly that it was not contrary to the Scriptures.
For people like Galileo and Kepler, faith and reason were the same thing. By definition, it's impossible to have faith that is not based on reason, nor is it possible to hold reason without faith. To do so is historical revisionism. If you have a bone to pick with faith, then the best place to start is with the life and death and resurrection of Christ in history. Did it happen? How soon after the events were the eye witness accounts recorded? At what cost? Independent? Do we read them as they were written? This is a matter of historicity: did it happen? Not of philosophical possibility (naturalism), or statistical possibility (frequentism).
What little I do know about many founders of the major fields of science seems to indicate that the founding men and women saw no such dichotomy. Or at the least were willing to struggle with the questions rather than throw the entire matter out as "unscientific."
Lord Kelvin and Michael Faraday seem to have both Christians and creationists. Newton was a Christian and claimed that "all my discoveries have been made in answer to prayer." Joule (the unit's namesake) was at least a theist. Then we have Louis Pasteur, Christopher Columbus, George Washington Carver, Samuel Morse, and Blaise Pascal - each "men of faith."
Not that the above list should be taken as an argument by authority for faith. Certainly not. However, it does leave me wondering how these great thinkers saw the world so differently.
David Seccombe, King of God's Kingdom, 280.
R. Hooykaas, Christian Faith and the Freedom of Science, 17. Encyclopedia Brittanica article: Kepler, Johannes.
Further, it's interesting that Kepler was schooled by a protege of Melanchthon, who was himself a key reformer with Luther and a contemporary of Tyndale. These men, together with Erasmus, were of the finest minds of their day, giants of the past, and giants still today.
"What little I do know about many founders of the major fields of science seems to indicate that the founding men and women saw no such dichotomy."
Yes, certainly "blind faith" and science are irreconcilable. Blind faith is a straw man. Men have wasted breath on it for years. Faith is quite different from blind faith. The two (faith and blind faith) are equally irreconcilable. Faith is simply acting today in the light of that which by reason one is fully persuaded of. It need not surprise that great men of reason and action are also men of deep, often Christian convictions about God.
"However, it does leave me wondering how these great thinkers saw the world so differently."
As John Newton wrote, "I once was blind but now I see". As Philip said to Nathanael, "come and see".
The Christian of the 1st century would convert from Judaism either after seeing the risen Jesus, as at least five hundred on one occasion in Jerusalem did at the time, or after hearing of him from those who had. Their witness, at pain of death, was based on what they had seen and heard. Former persecutors such as Saul of Tarsus, a protege of Gamaliel, converted after seeing the risen Christ for themselves. 
It is notable that at least two members of the Jewish ruling council, which tried Jesus, Nicodemus and Joseph of Arimathea, did not consent to the actions of the council and followed Christ. These men had education and wealth. They had access to Jesus and they believed, at great personal cost. Proclaiming a crucified criminal as Sovereign King, "God with us", would have been unthinkable for men such as these, were they not fully persuaded of God's raising him up, and of the entirety of the Old Testament Scriptures, one example being Isaiah 53, which pointed to a suffering and triumphant Messiah. "Oh death, where is thy sting? Oh grave, where is thy victory?" To a naturalistic worldview which claims the name of science whilst being far from it, the gospel is hard to accept. Is it true? Did it happen? In our ignorance, we think of God as silent, unyielding. God is not silent, he has spoken and revealed himself in history. Those with ears to hear, let them hear.
Today we see the Lord through the eyes of history, through the accounts of those who saw him. Our distance from the events need not be a hindrance, only the distance of the accounts from the events, and that is, in terms of history, a comparatively short span of only 20 to 60 years at most for the various accounts, today collectively known as the New Testament, the letters of Luke (a physician and meticulous historian in his own right by standards of ancient history), Peter, Paul, John, Matthew, Mark, James and Jude (brothers of Jesus) and outside of that, Tacitus, Suetonius, Josephus, Clement, Polycarp etc.
Concerning textual criticism of the New Testament documents, see:
F.F. Bruce, various works.
Paul Barnett, various works.
Sir Frederic Kenyon, director of the British Museum for 22 years, various works.
Note that the Wikipedia pages on these subjects are for once a poor source, they suffer from speculation, superstition, tradition, "sainthood", and the old school of form criticism ("Today it is no exaggeration to claim that a whole spectrum of main assumptions underlying Bultmann's Synoptic Tradition must be considered suspect." - Kelber, W.H.).
It is interesting that polemicists such as Richard Dawkins, Christopher Hitchens tend to avoid confronting the historicity of Christ, and Christ himself head on. They refuse to get into the ring, preferring easier prey. It is ironic that the brother of Christopher Hitchens, Peter Hitchens, a renowned journalist, would be an atheist who converted to Christ and has since written on the subject of "the rage against God".
 - "Many have undertaken to draw up an account of the things that have been fulfilled among us, just as they were handed down to us by those who from the first were eyewitnesses and servants of the word. With this in mind, since I myself have carefully investigated everything from the beginning, I too decided to write an orderly account for you, most excellent Theophilus, so that you may know the certainty of the things you have been taught." - Luke, Luke 1
 - "I passed on to you what was most important and what had also been passed on to me. Christ died for our sins, just as the Scriptures said. He was buried, and he was raised from the dead on the third day, just as the Scriptures said. He was seen by Peter and then by the Twelve. After that, he was seen by more than 500 of his followers at one time, most of whom are still alive, though some have died. Then he was seen by James and later by all the apostles. Last of all, as though I had been born at the wrong time, I also saw him. For I am the least of all the apostles. In fact, I'm not even worthy to be called an apostle after the way I persecuted God's church." - Paul, 1 Corinthians 15:3-9
 - "And if Christ has not been raised, our preaching is useless and so is your faith. More than that, we are then found to be false witnesses about God, for we have testified about God that he raised Christ from the dead. […] But Christ has indeed been raised from the dead, the first-fruits of those who have fallen asleep." - Paul, 1 Corinthians 15:20
 - "That which was from the beginning, which we have heard, which we have seen with our eyes, which we have looked at and our hands have touched — this we proclaim concerning the Word [the Logos, light, reason] of life. The life appeared; we have seen it and testify to it, and we proclaim to you the eternal life, which was with the Father and has appeared to us. We proclaim to you what we have seen and heard, so that you also may have fellowship with us. And our fellowship is with the Father and with his Son, Jesus Christ. We write this to make our joy complete." - John, 1 John 1
Biblical references Psalm 93:1, 96:10, and 1 Chronicles 16:30 include text stating that "the world is firmly established, it cannot be moved." In the same manner, Psalm 104:5 says, "the Lord set the earth on its foundations; it can never be moved." Further, Ecclesiastes 1:5 states that "And the sun rises and sets and returns to its place" etc. Certainly for the Biblical literalist there are places of direct disagreement.
"For people like Galileo and Kepler, faith and reason were the same thing."
I suppose one could have 'faith in reason' (that's a great oxymoron), and one could also perhaps 'reason in faith'. But at a deeper level, the two concepts are orthogonal and irreconcilable.
"By definition, it's impossible to have faith that is not based on reason, nor is it possible to hold reason without faith."
faith - Strong belief in God or in the doctrines of a religion, based on spiritual apprehension rather than proof.
reason - The power of comprehending, inferring, or thinking especially in orderly rational ways.
Here is the "Galileo affair" Wikipedia link for those who wish to read up on the matter: http://en.wikipedia.org/wiki/Galileo_affair
Your definition of faith mistakes faith for blind faith. Faith itself is based on reason, necessarily, at least that is the view throughout the Scriptures. You are wasting time on a straw man here. To say the two concepts of reason and faith are orthogonal is inaccurate. They go hand in hand. Or would you put yourself against giants?
"Some people, eager for money, have wandered from the faith and pierced themselves with many griefs." - Paul, in his letter to Timothy.
I won't pretend to fully understand quantum entanglement but jumping straight to the balls-out Infinite Multiverse Theory is a bit much. Occam's Razor, please.
Heliocentrism is and was obviously right, minus the blinders of religion. I don't think the situation re. many-worlds is even remotely comparable. Then again, if you have some compelling, credible yet accessible literature setting out the case to the contrary, why not share?
It is easy for us to judge now that we have pictures from space, but the concept would have been very non-intuitive in centuries past. For example, one does not feel the movement of the Earth.
"I don't think the situation re. many-worlds is even remotely comparable."
I agree that the comparison is not exact, yet it seems to be the best one available to support my stance. Both geocentrism and wave collapse were theories that appealed to commonsense notions but ignored irrefutable evidence. Geocentrists invented the preposterous epicycles, which supposedly allowed the planets to move "as if" they were orbiting the Sun. Single worlders invented wave collapse to allow photons to operate "as if" all possible paths were taken.
As far as literature goes, I'd recommend the two books by David Deutsch. He gets extremely ambitious at times, but I would classify the writing as "compelling, credible yet accessible literature".
EDIT: Wait, I just remembered one dual physics/philosophy major who believes in quantum collapse. But even she thinks that the Copenhagen Interpretation, the one you usually see in the media, is ridiculous.
I'll say that, personally, I find it much easier to make sense of situations involving long range entanglements when I think about them in terms of Many Worlds than when I think about them in terms of waveform collapse. And the term "teleportation" as applied to these results really does annoy me.
Many worlds is an interpretation. It is an untestable explanation of the evidence, there is no way to decide between many worlds and other models (besides, perhaps, convenience). It is not physics, but metaphysics.
I know some people are pushing many worlds with religious fervor and it is obnoxious. But as far as I know, despite the horde of internet acolytes that gives MW a weird thought-smell, it is still a mainstream interpretation and nobody is "prosecuted" for it.
Many Worlds is the theory that best fits the evidence of quantum experiments, keeping in mind Occam's razor. In fact, it is the only comprehensible one. Wave collapse is comparable to epicycles in that they are both attempts at harmonizing completely contradictory evidence with a preconceived common-sense assumption (single universe and geocentrism, respectively).
Aside from the fact this is factually incorrect, it's also a bad argument, as most people who propose paradigm-shifting theories are wrong.
Using this argument associates you with a lot of crackpots a lot more strongly than it associates you with Galileo.
I shall supply a snippet of the Galileo affair that corroborates my hand-wavy statement:
"One of the first suggestions of heresy that Galileo had to deal with came in 1613 from a professor of philosophy (what we would now call a professor of scientific theory), Cosimo Boscaglia, who was neither a theologian nor a priest. In conversation with Galileo's patron, Cosimo II de' Medici, Boscaglia gave the opinion that the telescopic discoveries were valid, but the motion of the Earth was obviously contrary to Scripture." http://en.wikipedia.org/wiki/Galileo_affair#Bible_argument
"...most people who propose paradigm-shifting theories are wrong."
Galileo's success worked against him - other paradigm-shifters were not legitimate threats to the status quo.