Is there a moduli space in semialgebraic geometry analogous to the Hilbert scheme in algebraic geometry?

The sort of thing I am imagining is an object in a category of semischemes: Ordinary schemes plus a subsheaf of semirings in the structure sheaf called the ``nonnegative functions.'' Presumably, in stalks, the nonnegative functions would be required to contain any square of any element of the structure sheaf.

If there is no development along these lines, then I would also be happy if anyone knows a good proxy that appears in the literature.

Is there a moduli space in semialgebraic geometry analogous to the Hilbert scheme in algebraic geometry?

The sort of thing I am imagining is an object in a category of semischemes:

Ordinary schemes plus a subsheaf of semirings in the structure sheaf called the "nonnegative functions." Presumably, in stalks, the nonnegative functions would be required to contain any square of any element of the structure sheaf.

If there is no development along these lines, then I would also be happy if anyone knows a good proxy that appears in the literature.