Set theoretical analogues of the Barwise-Schlipf theorem

We characterize nonstandard models of ZF (of arbitrary cardinality) that can be expanded to Goedel-Bernays class theory plus ${\mathrm{\Delta }}_1^1$-Comprehension. We also characterize countable nonstandard models of ZFC that can be expanded to Goedel-Bernays class theory plus ${\mathrm{\Sigma }}_1^1$-Choice.