  The mutual inclination between planets orbiting the same star provides key
information to understand the formation and evolution of multi-planet systems.
In this work, we investigate the potential of Gaia astrometry in detecting and
characterizing cold Jupiters in orbits exterior to the currently known TESS
planet candidates. According to our simulations, out of the $\sim 3350$ systems
expected to have cold Jupiter companions, Gaia, by its nominal 5-year mission,
should be able to detect $\sim 200$ cold Jupiters and measure the orbital
inclinations with a precision of $\sigma_{\cos i}<0.2$ in $\sim 120$ of them.
These numbers are estimated under the assumption that the orbital orientations
of the CJs follow an isotropic distribution, but these only vary slightly for
less broad distributions. We also discuss the prospects from radial velocity
follow-ups to better constrain the derived properties and provide a package to
do quick forecasts using our Fisher matrix analysis. Overall, our simulations
show that Gaia astrometry of cold Jupiters orbiting stars with TESS planets can
distinguish dynamically cold (mean mutual inclination $\lesssim5^\circ$) from
dynamically hot systems (mean mutual inclination $\gtrsim 20^\circ$), placing a
new set of constraints on their formation and evolution.
