value, while eq.  checks for parity asymmetry, as we will argue below. The above equations are motivated from the literature on binary mixtures and we provide a brief summary on the discovered ground state phases and some of their properties, which will be relevant in the following discussions.

For weak couplings there is a miscible phase $M$ with a high spatial overlap of the one-body densities $\rho_1^{\sigma}(z)$. As a result both components exhibit a Gaussian profile ($\Delta_{\sigma}=1$) and occupy the center of their trap ($d=0$). The state is disentangled and both species are condensed. For negative couplings, i.e. attractive interactions, the phase remains miscible and the widths of the Gaussian densities shrink with decreasing coupling strength. For stronger positive couplings three different phase separation scenarios are possible. In case the majority species occupies the trap center ($\Delta_{M}=1$), pushing the impurities outside in a way that the impurity density forms a shell around the majority density with two parity-symmetric humps ($\Delta_{I}<1$ and $d=0$), we have a core-shell $IMI$ phase. When the impurities remain at the trap center instead ($\Delta_{I}=1$) with the majority species forming a shell ($\Delta_{M}<1$ and $d=0$), we have a core-shell $MIM$ phase. Finally, when both species develop two 