Good question! It's dictatorship. In such a situation, any non-principal ultrafilter picks out one of the congruence classes and only listens to that one.

More generally, given any partition of an infinite set of voters into a finite disjoint union of sets, a non-principal ultrafilter picks out one member of the partition and only listens to that one. In other words, a non-principal ultrafilter disenfranchises arbitrarily large portions of the population. This is another reason it's not very useful for actually conducting elections!