  Recently, the magnetic topological insulator MnBi$_2$Te$_4$ emerged as a
competitive platform to realize quantum anomalous Hall (QAH) states. We report
a Berry-curvature splitting mechanism to realize the QAH effect in the
disordered magnetic TI multilayers when switching from an antiferromagnetic
order to a ferromagnetic order. We reveal that the splitting of spin-resolved
Berry curvature, originating from the separation of the mobility edge during
the magnetic switching, can give rise to a QAH insulator even \emph{without}
closing the band gap. We present a global phase diagram, and also provide a
phenomenological picture to elucidate the Berry curvature splitting mechanism
by the evolution of topological charges. At last, we predict that the Berry
curvature splitting mechanism will lead to a reentrant QAH effect, which can be
detected by tuning gate voltage. Our theory will be instructive for the studies
of the QAH effect in MnBi$_2$Te$_4$ in future experiments.
