  The increasingly significant tensions within $\Lambda$CDM, combined with the
lack of detection of dark matter (DM) in laboratory experiments, have boosted
interest in non-minimal dark sectors, which are theoretically well-motivated
and inspire new search strategies for DM. Here we consider, for the first time,
the possibility of DM having simultaneous interactions with photons, baryons,
and dark radiation (DR). We have developed a new and efficient version of the
Boltzmann code CLASS that allows for one DM species to have multiple
interaction channels. With this framework we reassess existing cosmological
bounds on the various interaction coefficients in multi-interacting DM
scenarios. We find no clear degeneracies between these different interactions
and show that their cosmological effects are largely additive. We further
investigate the possibility of these models to alleviate the cosmological
tensions, and find that the combination of DM-photon and DM-DR interactions can
at the same time reduce the $S_8$ tension (from $2.3\sigma$ to $1.2\sigma$) and
the $H_0$ tension (from $4.3\sigma$ to $3.1\sigma$). The public release of our
code will pave the way for the study of various rich dark sectors.
