A ganzstellensatz for semi-algebraic sets in real closed valued fields

Let $(K,\nu )$ be a real closed valued field, and let $S\subseteq K^n$ be a definable open semi-algebraic set. We find an algebraic characterization of rational functions which are OVF-integral on $S$. We apply the existing model theoretic framework for proving ganzstellens"atze, and need to control semi-sections and their relations to orderings. (joint work with Noa Lavi)