  A fusion category of rank $4$ has either four self-dual objects or exactly
two self-dual objects. We study fusion categories of rank $4$ with exactly two
self-dual objects, giving nearly a complete classification of those based ring
that admit pseudo-unitary categorification. More precisely, we show that if
$\mathcal{C}$ is such a fusion category, then its Grothendieck ring
$K(\mathcal{C})$ must be one of seven based rings, six of which have know
categorifications. In doing so, we classify all based rings associated with
near-group categories of the group $\mathbb{Z}/3\mathbb{Z}$.
