  Binaries comprised of a neutron star (NS) and a black hole (BH) have so far
eluded observations as pulsars and with gravitational waves (GWs). We model the
formation and evolution of these NS+BH binaries - including pulsar evolution -
using the binary population synthesis code COMPAS. We predict the presence of a
total of 50-2000 binaries containing a pulsar and a BH (PSR+BHs) in the
Galactic field. We find the population observable by the next generation of
radio telescopes, represented by the SKA and MeerKAT, current (LIGO/Virgo) and
future (LISA) GW detectors. We conclude that the SKA will observe 1-80 PSR+BHs,
with 0-4 binaries containing millisecond pulsars. MeerKAT is expected to
observe 0-40 PSR+BH systems. Future radio detections of NS+BHs will constrain
uncertain binary evolution processes such as BH natal kicks. We show that
systems in which the NS formed first (NSBH) can be distinguished from those
where the BH formed first (BHNS) by their pulsar and binary properties. We find
40% of the LIGO/Virgo observed NS+BHs from a Milky-Way like field population
will have a chirp mass $\geq 3.0$ M$_\odot$. We estimate the spin distributions
of NS+BHs with two models for the spins of BHs. The remnants of BHNS mergers
will have a spin of $\sim$0.4, whilst NSBH merger remnants can have a spin of
$\sim$0.6 or $\sim$0.9 depending on the model for BH spins. We estimate that
approximately 25-1400 PSR+BHs will be radio alive whilst emitting GWs in the
LISA frequency band, raising the possibility of joint observation by the SKA
and LISA.
