So, admittedly, I left a lot of details out of constructing W and I'm not sure if it even works as a model of ZFC. But it's what I intuitively think of as a model of internal set theory -- you have your standard sets as ultrapowers, and your "external/arbitrary" nonstandard sets as arbitrary subsets of standard sets. I'd be very curious to look at actually rigorous constructions of models of IST, which should provide the desired example.
9/9, fin.