If $T$ has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion $\mathscr{Q}$ such that, in any $\mathscr{Q}$-generic extension of the universe, there are non-isomorphic models $M_1$ and $M_2$ of $T$ that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if `c.c.c.' is replaced by other cardinal-preserving adjectives. We also give an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings.