This type of language can be confusing. It is not a "real" magnetic monopole but an "emergent" one. The collective behavior of all of the spins in the material shows the experimental signatures of a magnetic monopole. The idea is pretty simple to visualize - if you take a normal solenoid, the magnetic flux lines come out of one end and return into the other end. If you take the solenoid and stretch it into an infinitely long infinitesimally thin string the magnetic flux coming out of one end looks exactly like the magnetic field of a monopole. This is an old idea called the "dirac string". In these spin ice materials the magnetic moments of the rare earth atoms in the crystal are "geometrically frustrated" and cannot become all aligned or anti-aligned as in a normal magnetic material - so they apparently arrange themselves into strings, and the ends of the strings look like monopoles. (note that a line of tiny dipoles has approximately the same field as a solenoid)
Here are some other simple examples of particles with "emergent" properties:
(-) The conduction electrons in metals interact with the ions in the lattice as well as with each other through electrostatic forces, nonetheless they can be approximately described as a gas of non interacting electrons with a different "effective mass". In some cases (e.g. graphene, bismuth) the "effective mass" can be zero, in other cases (e.g. actinide compounds) they can be nearly as heavy as a proton. Of course the electron mass is still ~10^-30 kg
(-) In semiconductors, when an electron is removed from the filled valence band (e.g. by doping) it leaves behind a "hole" which behaves like a positively charged particle. negative charged electrons and positively charged holes can actually interact to form an "atom" called an exciton which shows a spectrum just like a hydrogen atom. This is a collective "emergent" behavior of the entire semiconductor - there aren't really positively charged particles or hydrogen-like atoms in the semiconductor.
They're not true magnetic monopoles. If I understand correctly, they're unusual crystal structures that approximate the behavior of a monopole in some ways, but they aren't true monopoles. Div(B) still equals zero.
Here's an article I found from a month ago that explains spin ice better than the OP:
What has been discovered are not of the exact sort of magnetic monopoles that Direc predicted, but these spin ice crystals have clearly divided north and south poles with measurable separation between the two poles.
Is there some kind of duality relationship between "proper point/small smear particles", and "emergent particle-like thingies dependent on the whole material's state"?
I.e. if we look at things one way, we get a set F of fundamental particles, and a set E of emergent particles.
But maybe there's some kind of transformation of the theory that takes E <-> F' and F <-> E'.
Can we "flip" the theory into another theory that reverses the notions "fundamental", and "emergent"?
Here are some other simple examples of particles with "emergent" properties:
(-) The conduction electrons in metals interact with the ions in the lattice as well as with each other through electrostatic forces, nonetheless they can be approximately described as a gas of non interacting electrons with a different "effective mass". In some cases (e.g. graphene, bismuth) the "effective mass" can be zero, in other cases (e.g. actinide compounds) they can be nearly as heavy as a proton. Of course the electron mass is still ~10^-30 kg
(-) In semiconductors, when an electron is removed from the filled valence band (e.g. by doping) it leaves behind a "hole" which behaves like a positively charged particle. negative charged electrons and positively charged holes can actually interact to form an "atom" called an exciton which shows a spectrum just like a hydrogen atom. This is a collective "emergent" behavior of the entire semiconductor - there aren't really positively charged particles or hydrogen-like atoms in the semiconductor.
there are many other more exotic examples