  Current searches for gravitational waves from compact-object binaries with
the LIGO and Virgo observatories employ waveform models with spins aligned (or
anti-aligned) with the orbital angular momentum. Here, we derive a new
statistic to search for compact objects carrying generic (precessing) spins.
Applying this statistic, we construct banks of both aligned- and generic-spin
templates for binary black holes and neutron-star--black-hole binaries, and
compare the effectualness of these banks towards simulated populations of
generic-spin systems. We then use these banks in a pipeline analysis of
Gaussian noise to measure the increase in background incurred by using generic-
instead of aligned-spin banks. Although the generic-spin banks have a factor of
ten to twenty more templates than the aligned-spin banks, we find an overall
improvement in signal recovery at fixed false-alarm rate for systems with
high-mass ratio and highly precessing spins ---up to 60\% for
neutron-star--black-hole mergers. This gain in sensitivity comes at a small
loss of sensitivity ($\lesssim$4\%) for systems that are already well-covered
by aligned-spin templates. Since the observation of even a single binary merger
with misalinged spins could provide unique astrophysical insights into the
formation of these sources, we recommend that the method described here be
developed further to mount a viable search for generic-spin binary mergers in
LIGO/Virgo data.
