  We construct refined tropical enumerative genus zero invariants of toric
surfaces that specialize to the tropical descendant genus zero invariants
introduced by Markwig and Rau when the quantum parameter tends to $1$. In the
case of trivalent tropical curves our invariants turn to be the
Goettsche-Schroeter refined broccoli invariants. We show that this is the only
possible refinement of the Markwig-Rau descendant invariants that generalizes
the Goettsche-Schroeter refined broccoli invariants. We discuss also the
computational aspect (a lattice path algorithm) and exhibit some examples.
