The merger of a black hole (BH) and a neutron star (NS) in most cases is expected to leave no material around the remnant BH; therefore, such events are often considered as sources of gravitational waves without electromagnetic counterparts. However, a bright counterpart can emerge if the NS is strongly magnetized, as its external magnetosphere can experience radiative shocks and magnetic reconnection during/after the merger. We use magnetohydrodynamic simulations in the dynamical spacetime of a merging BH–NS binary to investigate its magnetospheric dynamics. We find that compressive waves excited in the magnetosphere develop into monster shocks as they propagate outward. After swallowing the NS, the BH acquires a magnetosphere that quickly evolves into a split-monopole configuration and then undergoes an exponential decay (balding), enabled by magnetic reconnection and also assisted by the ringdown of the remnant BH. This spinning BH drags the split monopole into rotation, forming a transient pulsar-like state. It emits a striped wind if the swallowed magnetic-dipole moment is inclined to the spin axis. We predict two types of transients from this scenario: (1) a fast radio burst emitted by the shocks as they expand to large radii; and (2) an X-ray/γ-ray burst emitted by the e± outflow heated by magnetic dissipation.