  Recent experiments on the twisted semiconductor bilayer system $t$MoTe$_2$
have observed integer and fractional quantum anomalous Hall effects, which
occur in topological moir\'e bands at zero magnetic field. Here, we present a
global phase diagram of $t$MoTe$_2$ throughout the filling range $0< n\leq 1$
substantiated by exact diagonalization calculations. At a magic angle, we find
that the system resembles the lowest Landau level (LLL) to a remarkable degree,
exhibiting an abundance of incompressible fractional quantum anomalous Hall
states and compressible anomalous composite Fermi liquid states. Away from the
magic angle, particle-hole symmetry is strongly broken. Some LLL-like features
remain robust near half-filling, while others are replaced, predominantly by
charge density waves near $n=0$ and anomalous Hall Fermi liquids near $n=1$.
Among LLL-like phases, we find the anomalous composite Fermi liquid at
$n=\frac{1}{2}$ to be most robust against deviations from the magic angle.
Within the band-projected model, we show that strong particle-hole asymmetry
above the magic angle results from interaction-enhanced quasiparticle
dispersion near $n=1$. Our work sets the stage for future exploration of
LLL-like and beyond-LLL phases in fractional quantum anomalous Hall systems.
