  We explore whether non-standard dark sector physics might be required to
solve the existing cosmological tensions. The properties we consider in
combination are an interaction between the dark matter and dark energy
components, and a dark energy equation of state $w$ different from that of the
canonical cosmological constant $w=-1$. In principle, these two parameters are
independent. In practice, to avoid early-time, superhorizon instabilities,
their allowed parameter spaces are correlated. We analyze three classes of
extended interacting dark energy models in light of the 2019 Planck CMB results
and Cepheid-calibrated local distance ladder $H_0$ measurements of Riess et al.
(R19), as well as recent BAO and SNeIa distance data. We find that in
quintessence coupled dark energy models, where $w > -1$, the evidence for a
non-zero coupling between the two dark sectors can surpass the $5\sigma$
significance. On the other hand, in phantom coupled dark energy models, there
is no such preference for a non-zero dark sector coupling. All the models we
consider significantly raise the value of the Hubble constant easing the $H_0$
tension. The addition of low-redshift BAO and SNeIa measurements leaves some
residual tension with R19 but at a level that could be justified by a
statistical fluctuation. Bayesian evidence considerations mildly disfavour both
the coupled quintessence and phantom models, while mildly favouring a coupled
vacuum scenario, even when late-time datasets are considered. We conclude that
non-minimal dark energy cosmologies, such as coupled quintessence, phantom, or
vacuum models, are still an interesting route towards softening existing
cosmological tensions, even when low-redshift datasets and Bayesian evidence
considerations are taken into account. (abstract severely abridged)
