We switch off intra-component interactions to reduce the complexity and gain a better understanding of the separation process. A very rich phase diagram is revealed admitting two tri-critical points, where three phases may coexist. Second, within the framework of a mean-field approximation, we perform a detailed analysis of the separation mechanism. Equipped with this knowledge we derive a selection rule for phase separation processes and a simple algorithm to estimate the miscible-immiscible phase boundary. Finally, we investigate the deviations of the mean-field picture to a many-body approach. For this we use the Multi-Layer Multi-Configuration Time-Dependent Hartree method for bosonic mixtures [@MLX2013; @MLX2013b; @MLX2017]. We find that in the vicinity of the high-entanglement regime the phase diagram is indeed greatly affected. The symmetry-broken phase is replaced by the composite fermionization, while the onset of symmetry-breaking is linked to the degree of entanglement reaching a certain threshold. Furthermore, the location of this beyond mean-field regime strongly depends on the harmonic length scale ratio and the particle number ratio. We also find that the one-body density is in general not sufficient to distinguish between a core-shell phase and the composite fermionization.

This work is organized as follows. In Sec. \[sec:general\_setup\] we introduce our 