  The random interlacement point process (introduced by Sznitman, generalized
by Teixeira) is a Poisson point process on the space of labeled doubly infinite
nearest neighbour trajectories modulo time-shift on a transient graph $G$. We
show that the random interlacement point process on any transient transitive
graph $G$ is a factor of i.i.d., i.e., it can be constructed from a family of
i.i.d. random variables indexed by vertices of the graph via an equivariant
measurable map. Our proof uses a variant of the soft local time method
(introduced by Popov and Teixeira) to construct the interlacement point process
as the almost sure limit of a sequence of finite-length variants of the model
with increasing length. We also discuss a more direct method of proving that
the interlacement point process is a factor of i.i.d. which works if and only
if $G$ is non-unimodular.
