  Gravitational waves emitted by coalescing binary systems containing neutron
stars (or other compact objects) carry signatures of the stars' internal
equation of state, notably, through the influence of tidal deformations during
the binary's inspiral stage. While the leading-order tidal effects for
post-Newtonian binaries of compact bodies in general relativity are due to the
bodies' mass-quadrupole moments induced by gravitoelectric tidal fields, we
consider here the leading effects due to current-quadrupole moments induced by
gravitomagnetic tidal fields. We employ an effective action approach to
determine the near-zone gravitational field and the conservative orbital
dynamics, initially allowing for arbitrary (not just tidally induced)
current-quadrupoles; our approach significantly reduces the complexity of the
calculation compared to previous derivations of the conservative dynamics for
arbitrary multipoles. We finally compute the leading contributions from
gravitomagnetic tides to the phase and (for the first time) the mode amplitudes
of the gravitational waves from a quasi-circular binary inspiral, given in
terms of the bodies' quadrupolar gravitomagnetic tidal Love numbers (tidal
linear response coefficients in an adiabatic approximation). In the phase,
gravitomagnetic tides are suppressed by one post-Newtonian order relative to
gravitoelectric ones, but this is not always the case for the mode amplitudes.
In the $(\ell,|m|)=(2,1)$ and $(3,2)$ modes for example, they appear at the
same leading orders.
