There's no actual system that satisfies the axioms of the reals, but there (logically) could be. If you like, you could say that there is a "possible system" that satisfies those axioms (as long as they're not contradictory!).

The real answer is that talk of numbers as entities can be thought of as syntactic sugar for saying that certain logical implications hold. It's somewhat revisionary, in that that's not what people think that they are doing, and people talked about numbers long before they knew of any axiomatizations for them, but if you think about it it's pretty clear why those ways of talking would have worked, even if people hadn't quite figured out the right way to think about it yet.

If you like, you can think of it as saying: "Numbers don't exist as floaty entities, so strictly speaking normal number talk is all wrong. However, [facts about logical implications] are true, and there's a pretty clear truth-preserving mapping between the two, so perhaps this is what people were trying to get at."