  We perform hydrodynamic simulations of mass transfer in binaries that contain
a white dwarf and a neutron star (WD-NS binaries), and measure the specific
angular momentum of material lost from the binary in disc winds. By
incorporating our results within a long-term evolution model we measure the
long-term stability of mass transfer in these binaries. We find that only
binaries containing helium white dwarfs with masses less than a critical mass
of $M_{\rm WD,crit}=0.2\,M_\odot$ undergo stable mass transfer and evolve into
ultra-compact $X$-ray binaries. Systems with higher-mass white dwarfs
experience unstable mass transfer, which leads to tidal disruption of the white
dwarf. Our low critical mass compared to the standard jet-only model of mass
loss arises from the efficient removal of angular momentum in the mechanical
disc winds which develop at highly super-Eddington mass-transfer rates. We find
that the eccentricities expected for WD-NS binaries when they come into contact
do not affect the loss of angular momentum, and can only affect the long-term
evolution if they change on shorter timescales than the mass-transfer rate. Our
results are broadly consistent with the observed numbers of both ultra-compact
$X$-ray binaries and radio pulsars with white dwarf companions. The observed
calcium-rich gap transients are consistent with the merger rate of unstable
systems with higher-mass white dwarfs.
