Spectroscopic parameters and widths of the tensor hybrid mesons $H_{\mathrm{bc}}$ and ${\tilde{H}}_{\mathrm{bc}}$ with the structure $\bar{b}gc$ and spin-parities $J^{\mathrm{P}}=2^{-}$ and $J^{\mathrm{P}}=2^{+}$ are calculated with high accuracy in the QCD sum rule framework. Information on their masses $m=(7.214\pm 0.075)\mathrm{GeV}$ and $\tilde{m}=(7.685\pm 0.040)\mathrm{GeV}$ enable us to determine decay channels of $H_{\mathrm{bc}}$ and ${\tilde{H}}_{\mathrm{bc}}$. The full width of the meson $H_{\mathrm{bc}}$ is estimated by considering the decays $H_{\mathrm{bc}}\to D^{+}{\bar{B}}^{⁎0}$ and $D^0{B}^{⁎+}$. The channels ${\tilde{H}}_{\mathrm{bc}}\to B^{+}{D}^0$, $B^0{D}^{+}$, $B^{⁎+}{D}^{⁎0}$, $B^{⁎0}{D}^{⁎+}$, $B_s^0{D}_s^{+}$, and $B_s^{⁎0}{D}_s^{⁎+}$ are studied to find the width of the ${\tilde{H}}_{\mathrm{bc}}$ state. The widths of these processes are calculated using QCD three-point sum rule approach, which require estimation of the strong couplings at hybrid-meson-meson vertices. The results $(50.8\pm 9.8)\mathrm{MeV}$ and $(184.3\pm 22.8)\mathrm{MeV}$ for the full widths of the hybrid mesons $H_{\mathrm{bc}}$ and ${\tilde{H}}_{\mathrm{bc}}$ characterize them as relatively narrow and broad structures, respectively.