Let R be a real closed field, and let U be a semialgebraic subset of $R^n$. Let $S^0(U)$ be the ring of continuous R-valued semialgebraic functions. Also let $\tilde{U}$ be the subset of Spec$_r (R[X_1, \ldots, X_n])$ corresponding to U.

What does the real spectrum of $S^0(U)$ look like? Is it related to $\tilde{U}$ in some way?