Neutrinos remain mysterious. As an example, enhanced self-interactions ($\nu \mathrm{SI}$), which would have broad implications, are allowed. At the high neutrino densities within core-collapse supernovae, $\nu \mathrm{SI}$ should be important, but robust observables have been lacking. We show that $\nu \mathrm{SI}$ make neutrinos form a tightly coupled fluid that expands under relativistic hydrodynamics. The outflow becomes either a burst or a steady-state wind; which occurs here is uncertain. Though the diffusive environment where neutrinos are produced may make a wind more likely, further work is needed to determine when each case is realized. In the burst-outflow case, $\nu \mathrm{SI}$ increase the duration of the neutrino signal, and even a simple analysis of SN 1987A data has powerful sensitivity. For the wind-outflow case, we outline several promising ideas that may lead to new observables. Combined, these results are important steps toward solving the 35-year-old puzzle of how $\nu \mathrm{SI}$ affect supernovae.