  If T has only countably many complete types, yet has a type of infinite
multiplicity then there is a ccc forcing notion Q such that, in any 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 ccc forcing. We give examples showing that
the hypothesis on the number of complete types is necessary and what happens if
`ccc' is replaced 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.
