Scattering is used to probe matter and its interactions in all areas of physics. In ultracold atomic gases, control over pairwise interactions enables us to investigate scattering in quantum many-body systems. Previous experiments on colliding Bose-Einstein condensates have revealed matter-wave interference, haloes of scattered atoms, four-wave mixing and correlations between counter-propagating pairs. However, a regime with strong stimulation of spontaneous collisions analogous to superradiance has proved elusive. In this regime, the collisions rapidly produce highly correlated states with macroscopic population. Here we find that runaway stimulated collisions in Bose-Einstein condensates with periodically modulated interaction strength cause the collective emission of matter-wave jets that resemble fireworks. Jets appear only above a threshold modulation amplitude and their correlations are invariant even when the number of ejected atoms grows exponentially. Hence, we show that the structures and atom occupancies of the jets stem from the quantum fluctuations of the condensate. Our findings demonstrate the conditions required for runaway stimulated collisions and reveal the quantum nature of matter-wave emission.