  We propose a geometric phase-resolved tunneling valley Hall effect based on
the coherent transmission through two combined electric barriers in
$\alpha-\mathcal{T}_3$ lattices. It is shown that the backreflected electrons
at the barrier interface may acquire a valley-dependent geometric phase. The
coherence of this geometric phase leads to the valley-dependent skew tunneling,
which is responsible for the transverse valley current with zero net charge. We
further demonstrate that this charge-neutral transverse valley Hall current can
be electrically controlled by the gate voltages applied across the two combined
barrier regions and is absent when the two barriers are of equal height. Our
work opens a new approach to generating the valley Hall effect, suggesting
potential applications for valleytronic devices.
