  We explore a self-interacting neutrino cosmology in which neutrinos
experience a delayed onset of free-streaming. We use the effective field theory
of large-scale structure (LSS) to model matter distribution on mildly
non-linear scales within the self-interacting neutrino cosmology for the first
time. We perform the first combined likelihood analysis of BOSS full-shape
galaxy clustering, weak lensing, and Lyman-$\alpha$ forest measurements,
together with the cosmic microwave background (CMB) data from Planck. We find
that the full data set strongly favors presence of a flavor-universal neutrino
self-interaction, with a characteristic energy scale of order $10$ MeV. The
preference is at the $\sim 5\sigma$ level and is primarily driven by the
Lyman-$\alpha$ forest measurements and, to a lesser extent, the weak lensing
data from DES. The self-interacting neutrino model eases both the Hubble
tension and the $S_8$ tension between different cosmological data sets, but it
does not resolve either. Finally, we note a preference for a non-zero sum of
neutrino masses at the level of $\sim 0.3$ eV under this model, consistent with
previous bounds. These results call for further investigation in several
directions, and may have significant implications for neutrino physics and for
future new-physics searches with galaxy surveys.
