  In 2015, G.~Mikhalkin introduced a refined count for real rational curves in
toric surfaces. The counted curves have to pass through some real and complex
points located on the toric boundary of the surface, and the count is refined
according to the value of a so called quantum index. This count happens only to
depend on the number of complex points on each toric divisors, leading to an
invariant. First, we give a way to compute the quantum index of any oriented
real rational curve, getting rid of the previously needed "purely imaginary"
assumption on the complex points. Then, we use the tropical geometry approach
to relate these classical refined invariants to tropical refined invariants,
defined using Block-G\"ottsche multiplicity. This generalizes the result of
Mikhalkin relating both invariants in the case where all the points are real,
and the result of the author where complex points are located on a single toric
divisor.
