where the density of the two species are spatially separated into two domains. The phase separated MI phases, on the other hand, has random filling of the two species and are not separated into two distinct domains. The inclusion of the NN interactions modify the density distribution of the MI phases in the phase separated domain. To study this, we solve the Eq. (\[bhm\_ext\_ss\]) with finite $V_{12}$ and keep it fixed to a value of $0.05U$. We, then, increase the inter-species interaction $U_{12}$ from the miscible domain $U_{12}^2 < U_{11}U_{22}$ to the immiscible domain $U_{12}^2 > U_{11}U_{22}$. The phase diagrams for selected values of $U_{12}$ are shown in the Fig. \[ebhm\_v12\_0p05\].

### Miscible phase

In the miscible domain, $U_{12}^2 < U_{11}U_{22}$, the phase diagram has lobes of incompressible quantum phases having $\rho=n$ with $n\in \mathbb{N}$. These lobes are similar to those in the TBHM. In the present case, however, the $\rho=n$ lobes are intervened by lobes of DW quantum phases with half-integer total average occupancies $\rho = (2m+1)/2$ with $m\in \{0,\mathbb{N}\}$. The total occupancy $n_{p,q} = n^1_{p,q} + n^2_{p,q}$ of these phases have diagonal long-range order. This is essentially induced by the non-zero inter-species NN interaction, $V_{12}>0$. The particle densities 