I think this is a misunderstanding of what is eliminated in eliminative materialism, at least in a sensible eliminative materialism. I no longer remember Churchland's view precisely enough--maybe I should reread--to convey it rather than my internalized perspective which at the time I thought was not meaningfully different than hers.

What is eliminated isn't the phenomenon in question. The phenomenon is there, and is regular. What is eliminated is a separate existence.

There are various ways to axiomatize the natural numbers--Peano's for instance--but regardless of how you build them, once you form them into a ring with (0, 1, +, *), the prime numbers emerge. It is indeed the case that prime numbers, unlike most others, have no a, b > 1 such that a*b = p.

Now, you could be a mathematical realist, in which case you might want to consider that the prime numbers have their own existence, or a mathematical formalist, in which case you think "existence" is a different sort of thing than what prime numbers have (along with numbers at all); but you still get prime numbers, and there seems no option not to: you can't build a model of a ring on the integers for which prime numbers don't exist.

All eliminative materialism eliminates, in my understanding, is the possibility of failing to accrue the emergent properties while leaving the microscopic or more fundamental properties intact. They are, in this view, a logical consequence only, just like prime numbers are a logical consequence of having a ring over the integers. If I am mistaken, then I maintain that this is what it should be.

The primes are still there, the waves are still there, the consciousnesses are still there (perhaps). Because primes can at least be argued to have no independent existence, nothing "happens" because of primes themselves (though you may have physical consequences relating to prime numbers of physical objects). But the constituents of waves do have independent existence, so the waves themselves also have independent existence. It cannot be any other way: they are a necessary consequence of the constituents, so don't have an uncontingent existence, but there are indeed physical impacts of this emergent property (like erosion of coastal bluffs).

None of this gets at consciousness specifically--I'll save my answer to that for the other branch of the discussion, which involves whether consciousness specifically needs a strong emergence. But as far as emergence goes, I don't think it's fundamentally a different question than whether primes need to have some sort of independent ontological existence or whether primality is merely a consequence of (N, 0, 1, +, *) and the properties those things entail.