Isomorphism property in nonstandard extensions of theZFC universe

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We study models ofHST (a nonstandard set theory which includes, in particular, the Replacement and Separation schemata ofZFC in the language containing the membership and standardness predicates, and Saturation for well-orderable families of internal sets). This theory admits an adequate formulation of the isomorphism propertyIP, which postulates that any two elementarily equivalent internally presented structures of a well-orderable language are isomorphic. We prove thatIP is independent ofHST (using the class of all sets constructible from internal sets) and consistent withHST (using generic extensions of a constructible model ofHST by a sufficient number of generic isomorphisms).