When soft matter is driven out of equilibrium its constituents interact via effective interactions that escape Newton’s action–reaction principle. Prominent examples include the hydrodynamic interactions between colloidal particles driven in viscous fluids, phoretic interactions between chemically active colloids, and quorum-sensing interactions in bacterial colonies. Despite a recent surge of interest in nonreciprocal physics, a fundamental question remains: do nonreciprocal interactions alter or strengthen the ordered phases of matter driven out of equilibrium? Here, through a combination of experiments and simulations, we show how nonreciprocal forces propel and fission dislocations formed in hydrodynamically driven Wigner crystals. We explain how dislocation motility results in the continuous reshaping of grain-boundary networks, and how their fission reaction melts driven crystals from their interfaces. Beyond the specifics of hydrodynamics, we argue theoretically that topological defects and nonreciprocal interactions should invariably conspire to deform and ultimately destroy crystals.