Am I right in thinking that a set theory $T$ may interpret a set theory $U$ while it is the case that $T$ has models which are not models of $U\ ?$ 


In particular, it seems to me that this can happen in case $M$ is a minimal model of $T$. 



A question on models and interpretations of set theories
ZFC can interpret ZFC + $V=L$ since the constructible sets in any model of ZFC form a model of ZFC + $V=L$. However, if there is a model of ZFC then there is a model of ZFC + $V\neq L$.


More strikingly, ZFC can interpret [Aczel's AFA](http://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axiom) but since the latter refutes the axiom of foundation, no model of ZFC is a model of AFA.


