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Since pulsars were discovered as emitters of bright coherent radio emission more than half a century ago, the cause of the emission has remained a mystery. In this Letter we demonstrate that coherent radiation can be directly generated in non-stationary pair plasma discharges which are responsible for filling the pulsar magnetosphere with plasma. By means of large-scale two-dimensional kinetic plasma simulations, we show that if pair creation is non-uniform across magnetic field lines, the screening of electric field by freshly produced pair plasma is accompanied by the emission of waves which are electromagnetic in nature. Using localized simulations of the screening process, we identify these waves as superluminal ordinary (O) modes, which should freely escape from the magnetosphere as the plasma density drops along the wave path. The spectrum of the waves is broadband and the frequency range is comparable to that of observed pulsar radio emission.