  Anomalous Hall effect (AHE) is a fundamental spin-dependent transport
property that is widely used in spintronics. It is generally expected that
currents carrying net spin polarization are required to drive the AHE. Here we
demonstrate that, in contrast to this common expectation, a spin-neutral
tunneling AHE (TAHE), i.e. a TAHE driven by spin-neutral currents, can be
realized in an antiferromagnetic (AFM) tunnel junction where an AFM electrode
with a non-spin-degenerate Fermi surface and a normal metal electrode are
separated by a non-magnetic barrier with strong spin-orbit coupling (SOC). The
symmetry mismatch between the AFM electrode and the SOC barrier results in an
asymmetric spin-dependent momentum filtering of the spin-neutral longitudinal
current generating the transverse Hall current in each electrode. We predict a
sizable spin-neutral TAHE in an AFM tunnel junction with a RuO$_{2}$-type AFM
electrode and a SnTe-type SOC barrier and show that the Hall currents are
reversible by the N\'eel vector switching. With the Hall angle being comparable
to that in conventional AHE bulk materials, the predicted spin-neutral TAHE can
be used for the N\'eel vector detection in antiferromagnetic spintronics.
