  In physically realistic scalar-field based dynamical dark energy models
(including, e.g., quintessence) one naturally expects the scalar field to
couple to the rest of the model's degrees of freedom. In particular, a coupling
to the electromagnetic sector leads to a time (redshift) dependence of the
fine-structure constant and a violation of the Weak Equivalence Principle. Here
we extend the previous Euclid forecast constraints on dark energy models to
this enlarged (but physically more realistic) parameter space, and forecast how
well Euclid, together with high-resolution spectroscopic data and local
experiments, can constrain these models. Our analysis combines simulated Euclid
data products with astrophysical measurements of the fine-structure constant,
$\alpha$, and local experimental constraints, and includes both parametric and
non-parametric methods. For the astrophysical measurements of $\alpha$ we
consider both the currently available data and a simulated dataset
representative of Extremely Large Telescope measurements and expected to be
available in the 2030s. Our parametric analysis shows that in the latter case
the inclusion of astrophysical and local data improves the Euclid dark energy
figure of merit by between $8\%$ and $26\%$, depending on the correct fiducial
model, with the improvements being larger in the null case where the fiducial
coupling to the electromagnetic sector is vanishing. These improvements would
be smaller with the current astrophysical data. Moreover, we illustrate how a
genetic algorithms based reconstruction provides a null test for the presence
of the coupling. Our results highlight the importance of complementing surveys
like Euclid with external data products, in order to accurately test the wider
parameter spaces of physically motivated paradigms.
