  A large anomalous Hall effect (AHE) has recently been observed in the
intercalated transition metal dichalcogenide (TMDC) Co$_{1/3}$NbS$_{2}$ below a
known magnetic phase transition at $T_N$ = 29 K. The spins in this material are
widely believed to order in a highly symmetric collinear antiferromagnetic
configuration, causing extensive debate about how reports of an AHE can be
reconciled with such a state. In this article, we address this controversy by
presenting new neutron diffraction data on single crystals of
Co$_{1/3}$NbS$_{2}$ and an analysis that implies that moments in this material
order into a non-collinear configuration, but one that maintains the same
refelction symmetries as the collinear phase. We present new transport and
magneto-optic Kerr measurements which show that AHE signatures persist below
$T_N$ to temperatures as low as $T$ = 5 K and firmly associate them with the
long-range antiferromagnetic order. Finally, we show that these AHE signatures
can be quantitatively reproduced by density functional theory (DFT)
calculations based on the lattice and spin state determined with neutron
diffraction. These combined findings establishes the veracity of the 'crystal
Hall effect' picture, which shows how such effects can emerge from the shape of
magnetic orbitals in compounds containing chiral lattice symmetry regardless of
the symmetry of the ordered spin configuration. These results illuminate a new
path for the discovery of anomalous Hall materials and motivate a targeted
study of the transport properties of intercalated TMDCs and other compounds
containing antiferromagnetic order and chiral lattice symmetry.
