  The inclusion of aligned-spin effects in gravitational-wave search pipelines
for neutron-star--black-hole binary coalescence has been shown to increase the
astrophysical reach with respect to search methods where spins are neglected
completely, under astrophysically reasonable assumptions about black-hole
spins. However, theoretical considerations and population synthesis models
suggest that many of these binaries may have a significant misalignment between
the black-hole spin and the orbital angular momentum, which could lead to
precession of the orbital plane during the inspiral and a consequent loss in
detection efficiency if precession is ignored. This work explores the effect of
spin misalignment on a search pipeline that completely neglects spin effects
and on a recently-developed pipeline that only includes aligned-spin effects.
Using synthetic but realistic data, which could reasonably represent the first
scientific runs of advanced-LIGO detectors, the relative sensitivities of both
pipelines are shown for different assumptions about black-hole spin magnitude
and alignment with the orbital angular momentum. Despite the inclusion of
aligned-spin effects, the loss in signal-to-noise ratio due to precession can
be as large as $40\%$, but this has a limited impact on the overall detection
rate: even if precession is a predominant feature of neutron-star--black-hole
binaries, an aligned-spin search pipeline can still detect at least half of the
signals compared to an idealized generic precessing search pipeline.
