  We explore an interacting dark matter (IDM) model that allows for a fraction
of dark matter (DM) to undergo velocity-independent scattering with baryons. In
this scenario, structure on small scales is suppressed relative to the cold DM
scenario. Using the effective field theory of large-scale structure, we perform
the first systematic analysis of BOSS full-shape galaxy clustering data for the
IDM scenario, and we find that this model alleviates the $S_8$ tension between
large-scale structure and Planck data. Adding the $S_8$ prior from DES to our
analysis further leads to a mild $\sim3\sigma$ preference for a non-vanishing
DM-baryon scattering cross-section, assuming $\sim 10\%$ of DM is interacting
and has a particle mass of 1 MeV. This result produces a modest $\sim 20$%
suppression of the linear power at $k\lesssim 1~h$/Mpc, consistent with other
small-scale structure observations. Similar scale-dependent power suppression
was previously shown to have the potential to resolve $S_8$ tension between
cosmological data sets. The validity of the specific IDM model explored here
will be critically tested with upcoming galaxy surveys at the interaction level
needed to alleviate the $S_8$ tension.
