  We consider a two-component immiscible Bose-Einstein condensate with
dominating intra-species repulsive density-density interactions. In the
ground-state phase of such a system only one condensates is present. This can
be viewed as a spontaneous breakdown of $\mathbb{Z}_2$ symmetry. We study the
phase diagram of the system at finite temperature beyond mean-field
approximation. In the absence of rotation, we show that the system undergoes a
first order phase transition from this ground state to a miscible two-component
normal fluid as temperature is increased. In the presence of rotation, the
system features a competition between vortex-vortex interaction and short range
density-density interactions. This leads to a rotation-driven "mixing" phase
transition in a spatially inhomogeneous state with additional broken
$\mathrm{U}(1)$ symmetry. Thermal fluctuations in this state lead to nematic
two-component sheets of vortex liquids. At sufficiently strong inter-component
interaction, we find that the superfluid and $\mathbb{Z}_2$ phase transitions
split. This results in the formation of an intermediate state which breaks only
$\mathbb{Z}_2$ symmetry. It represents two phase separated normal fluids with
density imbalance.
