Despite its remarkable success, the standard $\Lambda$CDM paradigm has been challenged lately by significant tensions between different datasets. This has boosted interest in non-minimal dark sectors, which are theoretically well-motivated and inspire new search strategies for dark matter (DM). With this in mind, we have developed a new and efficient version of the Boltzmann code CLASS that allows for one DM species to have simultaneous interactions with photons, baryons, and dark radiation. As a proof-of-principle of this code, we have reassessed existing cosmological bounds on the various interaction coefficients in multi-interacting DM scenarios, showing that these constraints are robust to the underlying cosmological model. We have also investigated the ability of these models to alleviate the $H_0$ and $S_8$ tensions, showing that the combination of DM-photons and DM - dark radiation interactions can alleviate both tensions. The upcoming public release of our code will pave the way for the study of various rich dark sectors.