Tale of stable interacting dark energy, observational signatures, and
  the $H_0$ tension
  We investigate the observational consequences of a novel class of stable
interacting dark energy (IDE) models, featuring interactions between dark
matter (DM) and dark energy (DE). In the first part of our work, we start by
considering two IDE models which are known to present early-time linear
perturbation instabilities. Applying a transformation depending on the dark
energy equation of state (EoS) to the DM-DE coupling, we then obtain two novel
stable IDE models. Subsequently, we derive robust and accurate constraints on
the parameters of these models, assuming a constant EoS $w_x$ for the DE fluid,
in light of some of the most recent publicly available cosmological data. These
include Cosmic Microwave Background (CMB) temperature and polarization
anisotropy measurements from the \textit{Planck} satellite, a selection of
Baryon Acoustic Oscillation measurements, Supernovae Type-Ia luminosity
distance measurements from the JLA sample, and measurements of the Hubble
parameter up to redshift $2$ from cosmic chronometers. Our analysis displays a
mild preference for the DE fluid residing in the phantom region ($w_x<-1$),
with significance up to 95\% confidence level, while we obtain new upper limits
on the coupling parameter between the dark components. The preference for a
phantom DE suggests a coupling function $Q<0$, thus a scenario where energy
flows from the DE to the DM. We also examine the possibility of addressing the
$H_0$ and $\sigma_8$ tensions, finding that only the former can be partially
alleviated. Finally, we perform a Bayesian model comparison analysis to
quantify the possible preference for the two IDE models against the standard
concordance $\Lambda$CDM model, finding that the latter is always preferred
with the strength of the evidence ranging from positive to very strong.
