In this work we derive constraints on interacting dark matter-dark radiation
models from a full-shape analysis of BOSS-DR12 galaxy clustering data, combined
with Planck legacy cosmic microwave background (CMB) and baryon acoustic
oscillation (BAO) measurements. We consider a set of models parameterized
within the effective theory of structure formation (ETHOS), quantifying the
lifting of the ${S}_{8}$ tension in view of KiDS weak-lensing results. The most
favorable scenarios point to a fraction $f\sim 10-100\mathrm{\%}$ of interacting dark
matter as well as a dark radiation temperature that is smaller by a factor
$\xi \sim 0.1-0.15$ compared to the CMB, leading to a reduction of the tension
to the $\sim 1\sigma $ level. The temperature dependence of the interaction rate
favored by relaxing the ${S}_{8}$ tension is realized for a weakly coupled unbroken
non-Abelian $SU(N)$ gauge interaction in the dark sector. To map our results
onto this $SU(N)$ model, we compute higher-order corrections due to Debye
screening. We find a lower bound ${\alpha}_{d}\equiv {g}_{d}^{2}/(4\pi )\gtrsim {10}^{-8}({10}^{-9})$ for dark matter mass $1000(1)$ GeV for relaxing the ${S}_{8}$ tension,
consistent with upper bounds from galaxy ellipticities and compatible with
self-interactions relevant for small-scale structure formation.