Then again, have you ever observed a non-computable number as a component of the value of a wave function in nature? Real numbers add a lot of cruft which can never be practically observed (due to not being computable).
One of the main reasons we lose things is that our results have been built upon the real numbers since they were easier to conceptualize and to work with, but it's possible that a lot can be recovered (maybe even everything we care about) using only the computable numbers. For instance, see https://en.wikipedia.org/wiki/Computable_analysis for some results in recovering analysis (limits, differentiation, integration, etc) using the computable numbers.
Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals.
I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.
If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.
It is not a question of whether they exist in nature, but rather whether they are the more superior technique, or not, to explain nature.