  We consider gravitational wave (GW) sources with an associated
electromagnetic (EM) counterpart, and analyze the time delay between both
signals in the presence of lensing. If GWs have wavelengths comparable to the
Schwarzschild radius of astrophysical lenses, they must be treated with wave
optics, whereas EM waves are typically well within the approximation of
geometric optics. With concrete examples, we confirm that the GW signal never
arrives before its EM counterpart, if both are emitted at the same time.
However, during the inspiral of a binary, peaks of the GW waveform can arrive
before their EM counterpart. We stress this is only an apparent superluminality
since the GW waveform is both distorted and further delayed with respect to
light. In any case, measuring the multi-messenger time delay and correctly
interpreting it has important implications for unveiling the distribution of
lenses, testing the nature of gravity, and probing the cosmological expansion
history.
