  In this paper we discuss the Steiner property for minimal clusters in the
plane with an anisotropic double density. This means that we consider the
classical isoperimetric problem for clusters, but volume and perimeter are
defined by using two densities. In particular, the perimeter density may also
depend on the direction of the normal vector. The classical "Steiner property"
for the Euclidean case (which corresponds to both densities being equal to $1$)
says that minimal clusters are made by finitely many ${\rm C}^{1,\gamma}$ arcs,
meeting in finitely many "triple points". We can show that this property holds
under very weak assumptions on the densities. In the parallel paper "On the
Steiner property for planar minimizing clusters. The isotropic case" we
consider the isotropic case, i.e., when the perimeter density does not depend
on the direction, which makes most of the construction much simpler. In
particular, in the present case the three arcs at triple points do not
necessarily meet with three angles of $120^\circ$, which is instead what
happens in the isotropic case.
