  We demonstrate via exact diagonalization that AA-stacked TMD homobilayers
host fractional quantum anomalous Hall (FQAH) states with fractionally
quantized Hall conductance at fractional fillings $n=\frac{1}{3},\,
\frac{2}{3}$ and zero magnetic field. While both states are most robust at
angles near $\theta\approx 2^{\circ}$, the $n=\frac{1}{3}$ state gives way to a
charge density wave with increasing twist angle whereas the $n=\frac{2}{3}$
state survives across a much broader range of twist angles. We show that the
competition between FQAH states and charge density wave or metallic phases is
primarily controlled by the wavefunctions and dispersion of the underlying
Chern band, respectively. Additionally, Ising ferromagnetism is found across a
broad range of fillings where the system is insulating or metallic alike. The
spin gap is enhanced at filling fractions where integer and fractional quantum
anomalous Hall states are formed.
