It might make no impact on calculations (I'm not really qualified to say that) but it seems like it would make a big impact on epistemology and determinism. I think?
I think you're right about that for determinism, but the acceptance of intuitionist math requires a different epistemology. It is literally a different set of assumptions on what is knowable and what is true.
None of our instruments are infinite precision, nor are very many physical constants, so it sort of makes sense that tossing out real numbers doesn’t make any difference.