  We show that for every countable group, any sequence of approximate
homomorphisms with values in permutations can be realized as the restriction of
a sofic approximation of an orbit equivalence relation. Moreover, this orbit
equivalence relation is uniquely determined by the invariant random subgroup of
the approximate homomorphism. We record applications of this result to recover
various known stability and conjugacy characterizations for almost
homomorphisms of amenable groups.
