McCarty = Eff

Recently in a discussion with J. Lipton there came up the question whether the full and faithful embedding of D. McCarty's realisability model of IZF into the effective topos Eff is actually an equivalence. More concretely that means whether any object of the effective topos is really isomorphic to an object of the McCarty topos. I think the problem is highly nontrivial as the equality predicate on an object of the McCarty topos is derived form its inductive presentation. I would be pleased in any comments. Thanks in advance, Thomas Streicher