  Antiferromagnetic (AFM) spintronics provides a route towards energy-efficient
and ultrafast device applications. Achieving anomalous valley Hall effect
(AVHE) in AFM monolayers is thus of considerable interest for both fundamental
condensed matter physics and device enginering. Here we propose a route to
achieve AVHE in A-type AFM insulator composed of vertically-stacked monolayer
quantum anomalous Hall insulator (QAHI) with strain and electric field
modulations. Uniaxial strain and electric field generate valley polarization
and spin splitting, respectively. Using first-principles calculations,
$\mathrm{Fe_2BrMgP}$ monolayer is predicted to be a prototype A-type AFM
hosting \emph{valley-polarized quantum spin Hall insulator} (VQSHI) in which
AVHE and quantum spin Hall effect (QSHE) are synergized in a single system. Our
findings reveal a route to achieve multiple Hall effects in 2D tetragonal AFM
monolayers.
