If T has only countably many complete types, yet has a type of infinit
e multiplicity then there is a c.c.c. forcing notion Q such that, in a
ny Q-generic extension of the universe, there are non-isomorphic model
s 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 comple
te types is necessary and what happens if 'c.c.c.' is replaced by othe
r 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.