  We explore the phase diagrams of moir\'e materials in search of a new class
of intervalley-coherent correlated insulating state: the Chern texture
insulator (CTI). This phase of matter, proposed in a companion paper, breaks
valley $U(1)$ symmetry in a nontrivial fashion wherein the valley order
parameter is forced to texture in momentum space as a consequence of band
topology. Using detailed Hartree-Fock studies, we establish that the CTI
emerges as an energetically competitive intermediate-coupling ground state in
several moir\'e systems which lack a twofold rotation symmetry that forbids the
single-particle topology essential to the formation of the CTI valley texture.
