How do we know that distant galaxies are not made up of antimatter?

I am assuming you mean galaxies.

And no, apparently not. The photons are the same. Though with good enough observations (which we are getting towards) we supposedly tell whether a galaxy has helium or antihelium.

I think not. Even what we think of as empty space is not actually empty. It has a smattering of hydrogen and helium atoms floating around in it, plus energized particles that have been emitted from high energy sources such as galactic cores, Supernovas, the ultra-high energy gas clouds surrounding Super Massive Black Holes, Gamma Ray Bursts, and QUASARS. All these particles and atoms are observable from Earth, and they are not antiparticles. Even if there were some magical transition zone with matter filled space on one side, and antimatter filled space on the other; the interface between them would be brilliantly apparent from earth, because matter and antimatter would constantly be colliding there and self extinguishing in a relatively massive (compared to the mass of each particle or atom) energy release. And if there were no wall of transition from matter to antimatter, then matter in the region around an antimatter galaxy would interact with the edges of the antimatter galaxy itself. We don’t see that happening.

You mean the galaxies that we have seen and taken photos of with Hubble? The very fact that we can see them and take photos of them means that the portion of them we see is not antimatter. We can even tell what some things are made up of by detecting the radiation, so no, it isn´t antimatter. We can see the galaxies and their form, so those aren´t antimatter.

Once upon a time, 13.7 billion years ago, there was no time and space. Out of singularity, everything emerged- the Big Bang- a huge inflation occurred. Whatever stuff might have been created out of nothing, might have consisted of almost equal amounts of matter and antimatter- they annihilated into energy- why matter was slightly more than antimatter is one of the greatest mystery. This remaining matter became atoms and stars and galaxies and also a lot of mysterious stuff called dark matter.
As space-time continued expanding (the expansion is accelerating because of another great mystery called dark energy), galaxies became distant from each other. Only the galaxies within a galaxy cluster remain close by- all other clusters move away from one another. Thus our universe today must be made up entirely of matter and not antimatter. (and been like that since a long long time )

Or maybe there was a duality during big bang- maybe it was polar- like two universes were created- a matter universe and an antimatter universe. Ours we call matter universe by default.

@cazzie Photons and antiphotonms are the same particle/wave. We would be able to see a block of antimatter with ordinary photons reflected

We don’t. Isolated antimatter galaxies would look the same to our telescopes. Current big bang theory suggests however that antimatter galaxies do not exist. There seems to be a tiny preference for matter, see

http://public.web.cern.ch/public/en/research/CPViolation-en.html

If an antimatter galaxy were not that isolated cosmic radiation would send loads of antiprotons and antihelium nuclei in all direction causing visible annihilation effects when contact with matter is made. None of our observations do suggest this.