  In this paper we introduce a set of sufficient criteria for the construction
of relative hemisystems of the Hermitian space $\mathrm{H}(3,q^2)$, unifying
all known infinite families. We use these conditions to provide new proofs of
the existence of the known infinite families of relative hemisystems. Reproving
these results has allowed us to find new relative hemisystems closely related
to an infinite family of Cossidente's, and develop techniques that are likely
to be useful in finding relative hemisystems in future.
