  We report theoretical results for the stability of half-quantum vortices
(HQVs) in the superfluid phases of $^3$He confined in highly anisotropic Nafen
aerogel. Superfluidity of $^3$He confined in Nafen is the realization of a
"nematic superfluid" with Cooper pairs condensed into a single p-wave orbital
aligned along the anisotropy axis of the Nafen aerogel. In addition to the
nematic phase, we predict a second "chiral" phase that onsets at a lower
transition temperature. This chiral phase spontaneously breaks time-reversal
symmetry and is a topological superfluid. Both superfluid phases are equal-spin
pairing condensates that host arrays of HQVs as equilibrium states of rotating
superfluid $^3$He. We present results for the structure of HQVs, including
magnetic and topological signatures of HQVs in both the nematic and chiral
phases of $^3$He-Nafen.
