  The jet photosphere has been proposed as the origin for the gamma-ray burst
(GRB) prompt emission. In many such models, characteristic features in the
spectra appear below the energy range of the $\textit{Fermi}$ GBM detectors, so
joint fits with X-ray data are important in order to assess the photospheric
scenario. Here we consider a particular photospheric model which assumes
localized subphotospheric dissipation by internal shocks in a non-magnetized
outflow. We investigate it using Bayesian inference and a sample of 8 GRBs with
known redshifts which are observed simultaneously with $\textit{Fermi}$ GBM and
$\textit{Swift}$ XRT. This provides us with an energy range of $0.3$~keV to
$40$~MeV and much tighter parameter constraints. We analyze 32 spectra and find
that 16 are well described by the model. We also find that the estimates of the
bulk Lorentz factor, $\Gamma$, and the fireball luminosity, $L_{0,52}$,
decrease while the fraction of dissipated energy, $\varepsilon_{\mathrm{d}}$,
increase in the joint fits compared to GBM only fits. These changes are caused
by a small excess of counts in the XRT data, relative to the model predictions
from fits to GBM only data. The fact that our limited implementation of the
physical scenario yields 50\% accepted spectra is promising, and we discuss
possible model revisions in the light of the new data. Specifically, we argue
that the inclusion of significant magnetization, as well as removing the
assumption of internal shocks, will provide better fits at low energies.
