  We compute the differential cross-section for inclusive prompt photon$+$quark
production in deeply inelastic scattering of electrons off nuclei at small $x$
($e+A$ DIS) in the framework of the Color Glass Condensate effective field
theory. The result is expressed as a convolution of the leading order (in the
strong coupling $\alpha_{\mathrm{s}}$) impact factor for the process and
universal dipole matrix elements, in the limit of hard photon transverse
momentum relative to the nuclear saturation scale $Q_{s,A}(x)$. We perform a
numerical study of this process for the kinematics of the Electron-Ion Collider
(EIC), exploring in particular the azimuthal angle correlations between the
final state photon and quark. We observe a systematic suppression and
broadening pattern of the back-to-back peak in the relative azimuthal angle
distribution, as the saturation scale is increased by replacing proton targets
with gold nuclei. Our results suggest that photon+jet final states in inclusive
$e+A$ DIS at high energies are in general a promising channel for exploring
gluon saturation that is complementary to inclusive and diffractive dijet
production. They also provide a sensitive empirical test of the universality of
dipole matrix elements when compared to identical measurements in
proton-nucleus collisions. However because photon+jet correlations at small $x$
in EIC kinematics require jet reconstruction at small $k_\perp$, it will be
important to study their feasibility relative to photon-hadron correlations.
