  We study a class of multi-orbital models based on those proposed by
Venderbos, Hu, and Kane which exhibit an interplay of topology, interactions,
and fermion incoherence. In the non-interacting limit, these models exhibit
trivial and Chern insulator phases with Chern number $C \geq 1$ bands as
determined by the relative angular momentum of the participating orbitals.
These quantum anomalous Hall insulator phases are separated by topological
transitions protected by crystalline rotation symmetry, featuring Dirac or
quadratic band-touching points. Here we study the impact of Sachdev-Ye-Kitaev
(SYK) type interactions on these lattice models. Given the random interactions,
these models display `average symmetries' upon disorder averaging, including a
charge conjugation symmetry, so they behave as interacting models in
topological class $\mathbf{D}$ enriched by crystalline rotation symmetry. The
phase diagram of this model features a non-Fermi liquid at high temperature and
an `exciton condensate' with nematic transport at low temperature. We present
results from the free-energy, spectral functions, and the anomalous Hall
resistivity as a function of temperature and tuning parameters. Our results are
broadly relevant to correlated topological matter in multiorbital systems, and
may also be viewed, with a suitable particle hole transformation, as an
exploration of strong interaction effects on mean-field topological
superconductors.
