  We show that if a non-amenable, quasi-transitive, unimodular graph $G$ has
all degrees even then it has a factor-of-iid balanced orientation, meaning each
vertex has equal in- and outdegree. This result involves extending earlier
spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of
quasi-transitive, unimodular graphs. As a consequence, we also obtain that when
$G$ is regular (of either odd or even degree) and bipartite, it has a
factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov
beyond transitive graphs.
