  We introduce a non-reversible generalization of the Vertex-Reinforced Jump
Process (VRJP), which we call *-Vertex-Reinforced Jump Process (*-VRJP). It can
be seen as the continuous-time counterpart of the *-Edge-Reinforced Random Walk
(*-ERRW) (see Bacallado (2009), Bacallado, Sabot and Tarr\`es (2020)), which is
itself a non-reversible, and in fact Yaglom reversible, generalization of the
original ERRW introduced by Coppersmith and Diaconis in 1986. In contrast to
the classical VRJP, the *-VRJP is not exchangeable after time change, which
leads to several difficulties and new phenomena. However, we show that with
some appropriate randomization of the initial local time, it becomes partially
exchangeable after time change. We provide a representation of the "randomized"
*-VRJP as mixture of Yaglom reversible Markov jump processes with an explicit
mixing measure, and we prove that the non-randomized *-VRJP can be written as a
mixture of conditioned Markov processes.
