Universe with only exactly two charges of the same sign

I am trying to understand the cosmological consequences of having a Universe with non-zero total charge. For example, an infinite, unbounded universe with a finite total charge and thus zero charge density, explaining why we do not observe the consequences of non-zero charge. A friend raised the following objection: in this case, two charges of the same sign (say, two electrons, not accounting for wave-particle duality) would still keep interacting with each other even at a long distance, thus observing that total charge is not zero.

Let’s posit that the universe is infinite and that initial conditions are only two charges of the same sign. They start accelerating in exact opposite directions. Could they converge to a state in which they cannot infer the existence of the other? We are at a bit of loss as we are not sure whether relativistic effects should be factored in.