  Large-scale structure surveys have reported measurements of the density of
matter, $\Omega_\mathrm{m}$, and the amplitude of clustering, $\sigma_8$, that
are in tension with the values inferred from observations of the cosmic
microwave background. While this may be a sign of new physics that slows the
growth of structure at late times, strong astrophysical feedback processes
could also be responsible. In this work, we argue that astrophysical processes
are not independent of cosmology and that their coupling naturally leads to
stronger baryonic feedback in cosmological models with suppressed structure
formation or when combined with a mechanism that removes dark matter from
halos. We illustrate this with two well-motivated extensions of the Standard
Model known to suppress structure formation: massive neutrinos and decaying
dark matter. Our results, based on the FLAMINGO suite of hydrodynamical
simulations, show that the combined effect of baryonic and non-baryonic
suppression mechanisms is greater than the sum of its parts, particularly for
decaying dark matter. We also show that the dependence of baryonic feedback on
cosmology can be modelled as a function of the ratio
$f_\mathrm{b}/c^2_\mathrm{v}\sim
f_\mathrm{b}/(\Omega_\mathrm{m}\sigma_8)^{1/4}$ of the universal baryon
fraction, $f_\mathrm{b}$, to a velocity-based definition of halo concentration,
$c^2_\mathrm{v}$, giving an accurate fitting formula for the baryonic
suppression of the matter power spectrum. Although the combination of baryonic
and non-baryonic suppression mechanisms can resolve the tension, the models
with neutrinos and decaying dark matter are challenged by constraints on the
expansion history.
