  There has been a significant interest in modifications of the standard
$\Lambda$ Cold Dark Matter ($\Lambda$CDM) cosmological model prompted by
tensions between certain datasets, most notably the Hubble tension. The
late-time modifications of the $\Lambda$CDM model can be parametrized by three
time-dependent functions describing the expansion history of the Universe and
gravitational effects on light and matter in the Large Scale Structure. We
perform the first joint Bayesian reconstruction of these three functions from a
combination of recent cosmological observations, utilizing a theory-informed
prior built on the general Horndeski class of scalar-tensor theories. This
reconstruction is interpreted in light of the well-known $H_0$, the $S_8$ and
the $A_L$ tensions. We identify the phenomenological features that alternative
theories would need to have in order to ease some of the tensions, and deduce
important constraints on broad classes of modified gravity models. Among other
things, our findings suggest that late-time dynamical dark energy and
modifications of gravity are not likely to offer a solution to the Hubble
tension, or simultaneously solve the $A_L$ and $S_8$ tensions.
