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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 RuO2-type AFM electrode and a SnTe-type SOC barrier and show that the Hall currents are reversible by the Néel 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 Néel vector detection in antiferromagnetic spintronics.