  We perform force-free simulations for a neutron star orbiting a black hole,
aiming at clarifying the main magnetosphere properties of such binaries towards
their innermost stable circular orbits. Several configurations are explored,
varying the orbital separation, the individual spins and misalignment angle
among the magnetic and orbital axes. We find significant electromagnetic
luminosities, $L\sim 10^{42-46} \, [B_{\rm pole}/ 10^{12}{\rm G}]^2 \, {\rm
erg/s}$ (depending on the specific setting), primarily powered by the orbital
kinetic energy, being about one order of magnitude higher than those expected
from unipolar induction. The systems typically develop current sheets that
extend to long distances following a spiral arm structure. The intense
curvature of the black hole produces extreme bending on a particular set of
magnetic field lines as it moves along the orbit, leading to magnetic
reconnections in the vicinity of the horizon. For the most symmetric scenario
(aligned cases), these reconnection events can release large-scale plasmoids
that carry the majority of the Poynting fluxes. On the other hand, for
misaligned cases, a larger fraction of the luminosity is instead carried
outwards by large-amplitude Alfv{\'e}n waves disturbances. We estimate possible
precursor electromagnetic emissions based on our numerical solutions, finding
radio signals as the most promising candidates to be detectable within
distances of $\lesssim 200$\,Mpc by forthcoming facilities like the Square
Kilometer Array.
