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We demonstrate that magnetospheric interactions between merging neutron stars (NSs) generate dual-jetted current outflows, analogous to the Alfvén wings observed during planetary interactions in the solar system. Using three-dimensional relativistic MHD simulations, we model the interaction as a conducting sphere moving through a highly magnetized plasma of the companion's magnetosphere. Unusually, the interaction operates in a regime that is relativistic yet sub-Alfvénic. Electromagnetic draping amplifies magnetic fields in a narrow layer near the stellar surface, leading to the generation of electric currents along the local magnetic field. The generation of beamed outflows enhances the instantaneous power of the pulsar-like radio and high-energy emission, produces spin/orbital modulations, and is likely to lead to observable precursor emission preceding the main gravitational wave event.