Potential isomorphism and semi-proper trees

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 notion of weakly semiproper 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 show that the existence of weakly semi-proper trees is consistent relative to ZFC by proving the existence of weakly semi-proper trees under certain cardinal arithmetic assumptions. We also prove the consistency of the nonexistence of weakly semi-proper trees assuming the consistency of some large cardinals. 1