  Despite the success of the standard $\Lambda$CDM model of cosmology, recent
data improvements have made tensions emerge between low- and high-redshift
observables, most importantly in determinations of the Hubble constant $H_0$
and the (rescaled) clustering amplitude $S_8$. The high-redshift data, from the
cosmic microwave background (CMB), crucially relies on recombination physics
for its interpretation. Here we study how small-scale baryon inhomogeneities
(i.e., clumping) can affect recombination and consider whether they can relieve
both the $H_0$ and $S_8$ tensions. Such small-scale clumping, which may be
caused by primordial magnetic fields or baryon isocurvature below kpc scales,
enhances the recombination rate even when averaged over larger scales, shifting
recombination to earlier times. We introduce a flexible clumping model,
parametrized via three spatial zones with free densities and volume fractions,
and use it to study the impact of clumping on CMB observables. We find that
increasing $H_0$ decreases both $\Omega_m$ and $S_8$, which alleviates the
$S_8$ tension. On the other hand, the shift in $\Omega_m$ is disfavored by the
low-$z$ baryon-acoustic-oscillations measurements. We find that the clumping
parameters that can change the CMB sound horizon enough to explain the $H_0$
tension also alter the damping tail, so they are disfavored by current Planck
2018 data. We test how the CMB damping-tail information rules out changes to
recombination by first removing $\ell>1000$ multipoles in Planck data, where we
find that clumping could resolve the $H_0$ tension. Furthermore, we make
predictions for future CMB experiments, as their improved damping-tail
precision can better constrain departures from standard recombination. Both the
Simons Observatory and CMB-S4 will provide decisive evidence for or against
clumping as a resolution to the $H_0$ tension.
