  We calculate azimuthal correlations between the exclusively produced vector
meson and the scattered electron in Deep Inelastic Scattering processes at the
future Electron-Ion Collider (EIC). We identify "kinematical" and "intrinsic"
contributions to these correlations, and show that the correlations are
sensitive to the non-trivial correlations in the gluon distribution of the
target. Realistic predictions at the EIC kinematics are provided using two
different approaches to describe the dipole-proton interaction at relatively
small $x$.
