  In arXiv:1505.04338(4), G. Mikhalkin introduced a refined count for the real
rational curves in a toric surface which pass through certain conjugation
invariant set of points on the toric boundary of the surface. Such a set
consists of real points and pairs of complex conjugated points. He then proved
that the result of this refined count depends only on the number of pairs of
complex conjugated points on each toric divisor. Using the tropical geometry
approach and the correspondence theorem, he managed to compute these invariants
if all the points of the configuration are real. In this paper we address the
case when the configuration contains some pairs of conjugated points, all
belonging to the same component of the toric boundary.
