  We present a systematic study of the dynamics and gravitational-wave emission
of head-on collisions of spinning vector boson stars, known as Proca stars. To
this aim we build a catalogue of about 800 numerical-relativity simulations of
such systems. We find that the wave-like nature of bosonic stars has a large
impact on the gravitational-wave emission. In particular, we show that the
initial relative phase $\Delta \epsilon =\epsilon_1-\epsilon_2$ of the two
complex fields forming the stars (or equivalently, the relative phase at
merger) strongly impacts both the emitted gravitational-wave energy and the
corresponding mode structure. This leads to a non-monotonic dependence of the
emission on the frequency of the secondary star $\omega_2$, for fixed frequency
$\omega_1$ of the primary. This phenomenology, which has not been found for the
case of black-hole mergers, reflects the distinct ability of the Proca field to
interact with itself in both constructive and destructive manners. We postulate
this may serve as a smoking gun to shed light on the possible existence of
these objects.
