In what sense does sunlight have any coherence length at all? I was under the impression that coherence length was essentially a function of how tightly controlled the wavelength is of a monochromatic source like a laser. Sunlight is completely incoherent.
I mean, the spectrum still has a width, it's just a very large width -- which is why the coherence time is very low.
In a more abstract way, you're right that we are reaching for a metaphor here: if a frequency distribution is sharply peaked around one frequency then we can roughly say that the light "is that frequency" but then modify this by saying that the bits that are off-frequency, while they are "also that frequency," become "phase-incoherent" after a characteristic time or distance. That is not actually correct; they are simply different frequencies -- but it makes the interpretation easier. That metaphor begins to break down when you've got such a wide spread of frequencies: red light can only "pretend to be" blue light for a vanishing period of time.
To put it more plainly, the phase-coherence time is the inverse of the spectral width in frequency-space; multiply it by $c$ to get a phase-coherence length. If you applied it to a truly monochromatic ideal source, you would find that the time and length are infinite -- it only makes sense for real distributions with normal spread. But that's all distributions, including blackbody distributions like sunlight -- it's just that those distributions have very large spread and therefore very low coherence lengths: but it's the same definition either way.
Coherence is defined as a correlation of the light with itself. In simpler terms "how similar it is to itself after letting it evolve for a bit". Sunlight has a lot of different wavelengths evolving in many different ways, therefore it is not too coherent compared to an ideal single wavelength laser which will be identical even after "evolving" for a while.
However, if you look at sunlight at a certain time and then look at it one femtosecond later, the second look will still be similar to the first look to some extent, therefore you would have to say that the light is somewhat coherent.
This is why sunlight is definitely NOT "completely incoherent".
In fact any thermal radiator (e.g. black body) will be coherent over short distances proportional to the temperature of the body. Effectively the light's phase doesn't (on average) change faster than the quantum of energy (h-Plank's constant) divided by the thermal energy of the body (kT) so the power of a particular wavelength averaged in your eye can still have significant interference effects for a high temperature radiator. The distance of coherence is then related to that coherence time times the speed of light (so redder-longer wavelengths are coherent for fewer wavelenghths).
That leads to an equation of L= c*h / 4kT where the Sun can be treated roughly as 6000K. This gives a coherence length measured in microns, which is conveniently several times the half wave is necessary for interference for visible light (0.4-0.7um full wave). It's pretty cool, because I think it means that if you evolved eyes to see light near the peak of your black body radiator then you will be able to see Newton Rings for internally reflective thin membranes sized close to that peak due to the coherence length (due to Wien's law).
Coherence ultimately has to do with photons interfering with themselves. Films and beamsplitters can “cut” a photon in two pieces in the sense that the photon takes both paths. Separately, the photon has an uncertainty in its energy which corresponds to a bandwidth and hence a coherence time. If the system conspires to bring the photon paths back together again, meaningful interference effects occur when the photon is destroyed (measured) provided the path difference is within the coherence time. This is relatively easy to do in a bulk optical instrument like a Michelson or Mach-Zender interferometer: just carefully adjust the two optical paths to be equal. Remarkably, if you sweep one path back and forth with respect to the other, while passing through the zero optical path difference point, and take the Fourier transform of the detected signal, an accurate depiction of the light source spectrum is produced.
