  We investigate an effective model of proximity modified graphene (or
symmetrylike materials) with broken time-reversal symmetry. We predict the
appearance of quantum anomalous Hall phases by computing bulk band gap and
Chern numbers for benchmark combinations of system parameters. Allowing for
staggered exchange field enables quantum anomalous Hall effect in flat graphene
with Chern number $C=1$. We explicitly show edge states in zigzag and armchair
nanoribbons and explore their localization behavior. Remarkably, the
combination of staggered intrinsic spin-orbit and uniform exchange coupling
gives topologically protected (unlike in time-reversal systems) pseudohelical
states, whose spin is opposite in opposite zigzag edges. Rotating the
magnetization from out of plane to in plane makes the system trivial, allowing
to control topological phase transitions. We also propose, using density
functional theory, a material platform---graphene on Ising antiferromagnet
MnPSe$_3$---to realize staggered exchange (pseudospin Zeeman) coupling.
