  We give a dynamical characterization of categorical Morita equivalence
between compact quantum groups. More precisely, by a Tannaka-Krein type
duality, a unital C*-algebra endowed with commuting actions of two compact
quantum groups corresponds to a bimodule category over their representation
categories. We show that this bimodule category is invertible if and only if
the actions are free, with finite dimensional fixed point algebras, which are
in duality as Frobenius algebras in an appropriate sense. This extends the
well-known characterization of monoidal equivalence in terms of bi-Hopf-Galois
objects.
