Azimuthal asymmetries in exclusive electroproduction of a real photon from a longitudinally polarized deuterium target are measured with respect to target polarization alone and with respect to target polarization combined with beam helicity and/or beam charge. The asymmetries appear in the distribution of the real photons in the azimuthal angle ϕ around the virtual photon direction, relative to the lepton scattering plane. The asymmetries arise from the deeply virtual Compton scattering process and its interference with the Bethe-Heitler process. The results for the beam-charge and beam-helicity asymmetries from a tensor polarized deuterium target with vanishing vector polarization are shown to be compatible with those from an unpolarized deuterium target, which is expected for incoherent scattering dominant at larger momentum transfer. Furthermore, the results for the single target-spin asymmetry and for the double-spin asymmetry are found to be compatible with the corresponding asymmetries previously measured on a hydrogen target. For coherent scattering on the deuteron at small momentum transfer to the target, these findings imply that the tensor contribution to the cross section is small. Furthermore, the tensor asymmetry is found to be compatible with zero.