  We extend the Kulkarni class of multivariate phase--type distributions in a
natural time--fractional way to construct a new class of multivariate
distributions with heavy-tailed Mittag-Leffler(ML)-distributed marginals. The
approach relies on assigning rewards to a non--Mar\-ko\-vi\-an jump process
with ML sojourn times. This new class complements an earlier multivariate ML
construction \cite{multiml} and in contrast to the former also allows for tail
dependence. We derive properties and characterizations of this class, and work
out some special cases that lead to explicit density representations.
