  Many models of dark matter have been proposed in attempt to ease the $S_8$
tension between weak lensing and CMB experiments. One such exciting possibility
is cannibalistic dark matter (CanDM), which has exothermal number-changing
interactions allowing it to stay warm far into its non-relativistic regime.
Here we investigate the cosmological implications of CanDM and how it impacts
CMB anisotropies and the matter power spectrum, by implementing the model
within a linear Einstein-Boltzmann solver. We show that CanDM suppresses the
small scale matter power spectrum in a way very similar to light Warm Dark
Matter or Hot Dark Matter. However, unlike in those models, the suppression may
happen while the CanDM model still remains compatible with CMB constraints. We
put strong constraints on the interaction strength of CanDM as a function of
its abundance for both constant and temperature-dependent thermally-averaged
cross sections. We find that the CanDM model can easily solve the $S_8$ tension
(but has no impact on the Hubble tension). Indeed, it can accommodate values of
$S_8$ of the order of 0.76 while being compatible with CMB+BAO data. However,
as long as the $S_8$ tension remains moderate, the overall $\chi^2$ improvement
is relatively small given the number of extra free parameters, and the CanDM
model is not significantly preferred.
