I also wanted to say that I certainly believe that you've done real math and got interesting results from the sort of model we were discussing above. It does seem like, as you say, a rich and natural theory.
That said I wanted to make another counter argument here. Particularly based on the meaning of the phrase "pick a random real number". To pick a number at random is a necessarily randomized process, however it should give a classical output. After picking the number I should be able to make two copies of it, put them on ships flying away from eachother at the speed of light, and both ships should be able to perform the same computation (without entangling particle pairs beforehand).
Moreover any reasonable representation of a real number should admit operations like "what is its n^th digit".
Under these two constraints, you would have to flip an infinite number of coins, and it would take infinite time and space to pick a real number. Or in other words, you can't really.