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Welschinger numbers count (in a weighted manner) real rational curves on a
toric Del Pezzo surface, e.g. $\mathbb{P}^2$, which pass through a generic
configuration $\mathcal{P}$ of conjugation invariant points. Although it
is possible to show the invariance of these numbers with methods from
symplectic geometry, i.e. that they do not depend on the choice of
$\mathcal{P}$, there was no proof in the tropical context yet when
$\mathcal{P}$ also contains pairs of complex conjugate points.
In this talk, I want to describe the tropical situation and I want to
introduce certain tropical curves, namely broccoli curves, whose count is
invariant. Under certain conditions, one then gets Welschinger invariants
what yields a tropical proof of the invariance of these numbers. If time
permits, I will also illustrate the case of broccoli curves of genus 1.
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