When hypothetical neutrino secret interactions (νSI) are large, they form a fluid in a supernova (SN) core, flow out with sonic speed, and stream away as a fireball. For the first time, we tackle the complete dynamical problem and solve all steps, systematically using relativistic hydrodynamics. The impact on SN physics and the neutrino signal is remarkably small. For complete thermalization within the fireball, the observable spectrum changes in a way that is independent of the coupling strength. One potentially large effect beyond our study is quick deleptonization if νSI violate lepton number. By present evidence, however, SN physics leaves open a large region in parameter space, where laboratory searches and future high-energy neutrino telescopes will probe νSI.