Isolated photons with high transverse energy have been studied in $ep$ deep inelastic scattering with the ZEUS detector at HERA, using and integrated luminosity of $326~ \mathrm{pb}^{-1}$. The kinematic region includes photon virtualities $10<Q^2<350~\mathrm{GeV}^2$. Photons with transverse energy $4<E_{\gamma}^{T}<15 \mathrm{\;}\mathrm{GeV}$ and pseudorapidity $−0.7<\eta_{\gamma}<0.9$ were measured with accompanying jets having transverse energy and pseudorapidity $2.5<E_{jet}^T<35\mathrm{\;GeV}$ and $−1.5<\eta_{jet}<1.8$. Differential cross sections are presented for several different correlated variables: the fraction of the incoming photon energy and momentum that is transferred to the photon and the jet, the fraction of proton energy taken by the parton that interacts with the photon, the azimuthal angle and the pseudorapidity difference between the prompt photon and the jet and between the prompt photon and the scattered electron. Comparisons with different theoretical calculations are made.