Interference of proximity-induced superconducting correlations in mesoscopic metallic rings is sensitive to the magnetic flux Φ inside these rings. This is the reason for magnetoconductance oscillations in such systems. We detected experimentally and explained theoretically a novel effect: the phase of these oscillations can switch between 0 and π depending on the resistance of intermetallic interfaces and temperature. The effect is due to a nontrivial interplay between the proximity-induced enhancement of the local conductivity and the proximity-induced suppression of the density of states at low energies.