The papers "Forcing isomorphism" and "Forcing isomorphism II" might be relevant.

Review for the first paper: As the authors explain in their introduction, for any theory T it is easy to find two nonisomorphic models of T that become isomorphic via a forcing which collapses cardinals. By contrast, if T is classifiable, two nonisomorphic models of T of cardinal less than lambda remain nonisomorphic after a forcing which preserves cardinals under lambda and adds no countable subset to lambda. In this paper they show that (i) the condition of preservation of cardinals is not sufficient in the result just stated, since the classifiable theory of countably many nested equivalence relations, with binary splitting, has nonisomorphic models which become isomorphic after a ccc-forcing; (ii) all nonclassifiable theories have such a pair of models. The basis of the argument is the construction of a family of trees which can be made isomorphic by forcing.

Review for the first paper: This is the second paper justifying Shelah's theory of classification, that is, showing that one cannot do better. As in Part I [J. T. Baldwin, M. C. Laskowski and S. Shelah, J. Symbolic Logic 58 (1993), no. 4, 1291–1301; MR1253923 (94m:03054)], only c.c.c. forcings, which preserve cardinalities and cofinalities, are considered. This time it is shown that, given a small superstable non-omega-stable theory T, there exists an extension by forcing of the universe with two nonisomorphic models of T, with the property that they can be made isomorphic by forcing; the question of whether such a pair of models must exist in the ground universe is left open. A corollary is that the invariants for the model of a non-omega-stable classifiable theory cannot be substantially simplified; in particular, the models of T cannot be classified by independent trees of finite subsets. In such a theory T there is a type p of infinite multiplicity; each of the two models constructed realises a suitably chosen generic subset of the set of strong types extending p. In a second step new automorphisms of this set are added to make the two models isomorphic. For this construction, a measure on this space of strong types is defined. The last section contains examples and counterexamples, and in particular one showing the fragility of the membership relation in a pseudo-elementary class.

The papers "Forcing isomorphism" and "Forcing isomorphism II" might be relevant.

Review for the first paper: As the authors explain in their introduction, for any theory $T$ it is easy to find two non-isomorphic models of $T$ that become isomorphic via a forcing which collapses cardinals. By contrast, if $T$ is classifiable, two non-isomorphic models of $T$ of cardinal less than lambda remain non-isomorphic after a forcing which preserves cardinals under lambda and adds no countable subset to lambda. In this paper they show that (i) the condition of preservation of cardinals is not sufficient in the result just stated, since the classifiable theory of countably many nested equivalence relations, with binary splitting, has non-isomorphic models which become isomorphic after a ccc-forcing; (ii) all non-classifiable theories have such a pair of models. The basis of the argument is the construction of a family of trees which can be made isomorphic by forcing.

Review for the second paper: This is the second paper justifying Shelah's theory of classification, that is, showing that one cannot do better. As in Part I [J. T. Baldwin, M. C. Laskowski and S. Shelah, J. Symbolic Logic 58 (1993), no. 4, 1291–1301; MR1253923 (94m:03054)], only c.c.c. forcings, which preserve cardinalities and cofinalities, are considered. This time it is shown that, given a small superstable non-$\omega$-stable theory $T$, there exists an extension by forcing of the universe with two non-isomorphic models of $T$, with the property that they can be made isomorphic by forcing; the question of whether such a pair of models must exist in the ground universe is left open. A corollary is that the invariants for the model of a non-$\omega$-stable classifiable theory cannot be substantially simplified; in particular, the models of $T$ cannot be classified by independent trees of finite subsets. In such a theory $T$ there is a type $p$ of infinite multiplicity; each of the two models constructed realizes a suitably chosen generic subset of the set of strong types extending $p$. In a second step new automorphisms of this set are added to make the two models isomorphic. For this construction, a measure on this space of strong types is defined. The last section contains examples and counterexamples, and in particular one showing the fragility of the membership relation in a pseudo-elementary class.