  We give a definition of tropical orbit spaces and their morphisms. We show
that, under certain conditions, the weighted number of preimages of a point in
the target of such a morphism does not depend on the choice of this point. We
equip the moduli spaces of elliptic tropical curves with a structure of
tropical orbit space and, using our results on tropical orbit spaces, simplify
the known proof of the fact that the weighted number of plane elliptic tropical
curves of degree d with fixed j-invariant which pass through 3d-1 points in
general position in $\RR^2$ is independent of the choice of a configuration of
points.
