  We study a notion of potential isomorphism, where two structures are said to
be potentially isomorphic if they are isomorphic in some generic extension that
preserves stationary sets and does not add new sets of cardinality less than
the cardinality of the models. We introduce the notions of semi-proper and
weakly semi-proper trees, and note that there is a strong connection between
the existence of potentially isomorphic models for a given complete theory and
the existence of weakly semi-proper trees. We prove the existence of
semi-proper trees under certain cardinal arithmetic assumptions. We also show
the consistency of the non-existence of weakly semi-proper trees assuming the
consistency of some large cardinals.
