>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable?

I guess the naive answer is completeness. Every Cauchy sequence converges to a member of the space. For example, quantum mechanics relies on the formalism of Hilbert space, defined as a complete inner product space. This gives us nice things like the spectral theorem for unbounded operators, without which we wouldn't be able to define probability (the Born rule) or time evolution (the operator exponential e^-iHt).

Can you formalize quantum mechanics using computable numbers? I don't actually know, but let's say yes. What benefit do we actually gain that we didn't have in standard functional analysis?