How does anti-matter relate to dark matter?

It doesn’t. Anti-matter is essentially the same as ordinary matter with its charge reversed. Otherwise, it acts exactly like ordinary matter.

Dark matter is a special form of ordinary matter that’s non-baryonic. This means that it doesn’t interact with the electromagnetic force. Light goes straight through it without stopping, slowing, or scattering. Magnetic fields don’t move it. Nothing EM can affect it in any way. I believe that it still interacts with the weak and strong forces at least somewhat (someone correct me if I’m wrong), though those are absolutely useless on astronomical scales. Thus, we’re left with gravity being the only significant force that affects dark matter.

My question is; how can we know that it’s nonbaryonic dark matter?

It’s fairly simple. Of the light matter, as you mentioned, you have either gas clouds or galaxies. The galaxies only make up about 10% of the light mass, and they’re relatively compact, so they made it through the collision pretty much without, erm, collision. The gas, though, collided a lot.

Now, the mass of the dark matter and the properties of the lensing indicate that there was a *lot* of it. There was definitely more dark matter than there was light gas. If it was baryonic, it would have interacted in the same fashion as the light gas – it would have collided and produced shockwaves. Since it didn’t, we can be fairly certain that it wasn’t baryonic.

Plus, baryonic matter interferes with light (because it interacts with the electromagnetic force). Clouds of gas dim or block out light, or they glow from within because they’ve got stars inside of them. However, the giant mass of dark matter isn’t interfering with light at all, except for the gravitational lensing. Thus, another indication that its nonbaryonic.

I understand how the results show that the majority of each cluster’s original mass did not interact during the collision. And given that much of this un-reactive mass can’t be seen, it must (by definition) be dark matter.

My question is; how can we know that it’s nonbaryonic dark matter?

The assumption here seems to be that all of the matter in the clusters was either hot X-ray emitting intergalactic gas, visible galaxies, or nonbaryonic dark matter. Isn’t it possible that the clusters originally consisted of baryonic dark matter which, like the galaxies, did not interact during the collision?

Presumably, the galaxies themselves are ordinary (baryonic) visible matter, and they did not interact during the collision. Couldn’t galaxy-sized “blobs” of baryonic dark matter also behave as non-interactive particles during the collision?