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.