The jet photosphere has been proposed as the origin for the gamma-ray burst (GRB) prompt emis-sion. In many such models, characteristic features in the spectra appear below the energy range of theFermiGBM detectors, so joint fits with X-ray data are important in order to assess the photosphericscenario. Here we consider a particular photospheric model which assumes localized subphotosphericdissipation by internal shocks in a non-magnetized outflow. We investigate it using Bayesian inferenceand a sample of 8 GRBs with known redshifts which are observed simultaneously withFermiGBM andSwiftXRT. This provides us with an energy range of 0.3 keV to 40 MeV and much tighter parameterconstraints. We analyze 32 spectra and find that 16 are well described by the model. We also findthat the estimates of the bulk Lorentz factor, Γ, and the fireball luminosity,L0,52, decrease while thefraction of dissipated energy,εd, increase in the joint fits compared to GBM only fits. These changesare caused by a small excess of counts in the XRT data, relative to the model predictions from fits toGBM only data. The fact that our limited implementation of the physical scenario yields 50% acceptedspectra 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 internalshocks, will provide better fits at low energies.