Universe depending on all real numbers would be uncomputable, because most real numbers are uncomputable.
This follows from the fact that set of all Turing machines is countable (you can write an algorithm that lists all Turing machines), but set of real numbers is uncountable. So, by Cantor's diagonal argument you can prove that there are real numbers that can not be computed.
My guess is that universe can not depend on real numbers (i.e. real numbers are not real) because real numbers introduce infinity related breakdowns in physical models. For example, if there exist a physical state can encode arbitrary real number then that would allow to store infinite amount of information. But it is not possible to prove that something does not exist, see arguments related to Russell's teapot.
https://en.wikipedia.org/wiki/Computable_number