We investigate Aharonov-Bohm oscillations of the current through a strongly correlated quantum dot embedded in an arbitrary scattering geometry. Resonant-tunneling processes lead to a flux-dependent renormalization of the dot level. As a consequence, we obtain a fine structure of the current oscillations, which is controlled by quantum fluctuations. Strong Coulomb repulsion leads to a continuous bias voltage dependent phase shift and, in the nonlinear response regime, destroys the symmetry of the differential conductance under a sign change of the external flux.