  Double Dirac fermions have recently been identified as possible
quasiparticles hosted by three-dimensional crystals with particular
non-symmorphic point group symmetries. Applying a combined approach of
ab-initio methods and dynamical mean field theory, we investigate how
interactions and double Dirac band topology conspire to form the electronic
quantum state of Bi$_2$CuO$_4$. We derive a downfolded eight-band model of the
pristine material at low energies around the Fermi level. By tuning the model
parameters from the free band structure to the realistic strongly correlated
regime, we find a persistence of the double Dirac dispersion until its
constituting time reveral symmetry is broken due to the onset of magnetic
ordering at the Mott transition. We analyze pressure as a promising route to
realize a double-Dirac metal in Bi$_2$CuO$_4$.
