  We study coherent diffractive photon and vector meson production in
electron-proton and electron-nucleus collisions within the Color Glass
Condensate effective field theory. We show that electron-photon and
electron-vector meson azimuthal angle correlations are sensitive to non-trivial
spatial correlations in the gluon distribution of the target, and perform
explicit calculations using spatially dependent McLerran-Venugopalan initial
color charge configurations coupled to the numerical solution of small $x$
JIMWLK evolution equations. We compute the cross-section differentially in
$Q^2$ and $|t|$ and find sizeable anisotropies in the electron-photon and
electron-$\mathrm{J}/\psi$ azimuthal correlations ($v_{1,2} \approx 2 - 10 \%$)
in electron-proton collisions for the kinematics of the future Electron-Ion
Collider. In electron-gold collisions these modulations are found to be
significantly smaller ($v_{1,2} <0.1 \%$). We also compute incoherent
diffractive production where we find that the azimuthal correlations are
sensitive to fluctuations of the gluon distribution in the target.
