We consider a phase-separating mixture of active and passive fluids and explore morphological asymmetries of the emerging dominantly bicontinous dynamic emulsion. Two-dimensional numerical simulations reveal that the geometric and topological asymmetries can solely be explained by self-organized flows in the active region. As in inertial turbulence an inverse energy cascade in the active region leads to the formation of condensates. The size of these mesocales vortices is determined by the locally available space in the emulsion. As these condensates accumulate energy they impact the fluctuation of the surrounding interface and thus form a tight coupling between the flow field and the dynamic morphology. While explored for active/passive systems the symmetry-breaking mechanism can be generalized to heterogeneous active systems and proposes a way to control the morphology of various functional soft materials.