  The deformation of a nonrotating body resulting from the application of a
tidal field is measured by two sets of Love numbers associated with the
gravitoelectric and gravitomagnetic pieces of the tidal field, respectively.
The gravitomagnetic Love numbers were previously computed for fluid bodies,
under the assumption that the fluid is in a strict hydrostatic equilibrium that
requires the complete absence of internal motions. A more realistic
configuration, however, is an irrotational state that establishes, in the
course of time, internal motions driven by the gravitomagnetic interaction. We
recompute the gravitomagnetic Love numbers for this irrotational state, and
show that they are dramatically different from those associated with the strict
hydrostatic equilibrium: While the Love numbers are positive in the case of
strict hydrostatic equilibrium, they are negative in the irrotational state.
Our computations are carried out in the context of perturbation theory in full
general relativity, and in a post-Newtonian approximation that reproduces the
behavior of the Love numbers when the body's compactness is small.
