  Given a semi-algebraic set S, we study compactifications of S that arise from
embeddings into complete toric varieties. This makes it possible to describe
the asymptotic growth of polynomial functions on S in terms of combinatorial
data. We extend our earlier work to compute the ring of bounded functions in
this setting and discuss applications to positive polynomials and the moment
problem. Complete results are obtained in special cases, like sets defined by
binomial inequalities. We also show that the wild behaviour of certain examples
constructed by Krug and by Mondal-Netzer cannot occur in a toric setting.
