Just as the gravitational attraction of large gas clouds isn't typically enough to bring them together in intergalactic space gravitational repulsion wouldn't be enough to keep them apart. Instead, ordinary kinetic gas dynamics is what governs a lot of their behavior. Which means that the intergalactic gas clouds from a matter galaxy cluster would definitely end up intermingling with other intergalactic gas clouds from a nearby anti-matter galaxy cluster. And when that happens, the electrostratic attraction of oppositely charged particles would result in a high rate of annihilations, enough to shine brighter than even many billions of stars.

Also, it wouldn't matter if the gravitational attraction between matter and anti-matter were negative, that would mean that instead of being pulled toward the most massive nearby collection of matter an anti-matter galaxy (or cluster) would be repulsed. But that would simply drive the anti-matter galaxy into another region filled with matter galaxies, just at a lower density. Your theory really only works if there are enormous divisions between matter and anti-matter sections across the entire visible Universe, which there's no evidence of whatsoever. Even an anti-matter galaxy hiding in a massive intergalactic void doesn't help much because even such voids are filled with low density gas and dust clouds.

If the gravitational force between the two is repulsive, wouldn't that imply negative mass because of what we know of the Higgs field? Then we'd be deep into exotic matter territory and theoretically, even a small amount can warp space time very differently than regular matter so at galactic scales, shouldn't we see some pretty extreme evidence of this?

Given how little we know about the Higgs -- namely, that we're pretty certain that at least one particle exists which probably behaves in the same manner as a Higgs boson would on several tests -- I'd hesitate to draw conclusions on the basis of how a particular model suggests that the Higgs field behaves.

Galaxies come into contact regularly though, and we don't have any evidence that they interact in such a way.

Galaxies which gravitationally attract come into contact regularly.