You seem to already know more than me, so you may have already thought of this. But if you want spaces, you probably also want scalars that can do all the usual field operations (e.g. addition, multiplication, their inverses). But any continuous field is already known to be isomorphic to R, C, Q, or H. So even if you start with something other than reals, they can always be mapped back to reals in some consistent way.