  We perform magnetohydrodynamic simulations in full general relativity (GRMHD)
of a binary black hole-neutron star on a quasicircular orbit that undergoes
merger. The binary mass ratio is 3:1, the black hole initial spin parameter
$a/m=0.75$ ($m$ is the black hole Christodoulou mass) aligned with the orbital
angular momentum, and the neutron star is an irrotational $\Gamma=2$ polytrope.
About two orbits prior to merger (at time $t=t_B$), we seed the neutron star
with a dynamically weak interior dipole magnetic field that extends into the
stellar exterior. At $t=t_B$ the exterior has a low-density atmosphere with
constant plasma parameter $\beta\equiv P_{\rm gas}/P_{\rm
  mag}$. Varying $\beta$ at $t_B$ in the exterior from $0.1$ to $0.01$, we find
that at a time $\sim 4000{\rm M} \sim 100(M_{\rm
  NS}/1.4M_\odot)$ms [M is the total (ADM) mass] following the onset of
accretion of tidally disrupted debris, magnetic winding above the remnant black
hole poles builds up the magnetic field sufficiently to launch a mildly
relativistic, collimated outflow - an incipient jet. The duration of the
accretion and the lifetime of the jet is $\Delta t\sim 0.5(M_{\rm
NS}/1.4M_\odot)$s. Our simulations furnish the first explicit examples in GRMHD
which show that a jet can emerge following a black hole - neutron star merger.
