Real enumerative invariants

In the lecture I gave at the AIM workshop on Amoebas and tropical geometry, I defined some enumerative invariants of real algebraic convex 3-manifolds. For example, through a generic configuration of $2d$ real points in the complex projective space, there passes only finitely many irreducible real rational curves of degree $d$. Their real parts provide a collection of embedded knots in $\mathbb{RP}^3$. Equip this real projective space with a spin structure. Then it is possible to define a spinor orientation on these knots. Indeed, considering a real subholomorphic line bundle of maximal degree in the normal bundle of the curves, one first defines a framing on these knots. From this framing, one can then build a loop in the $SO_3 (\mathbb{R})$-principal bundle of orthonormal frames of $\mathbb{RP}^3$. Then, the spinor orientation of the real curve is the obstruction to lift this loop as a loop of the Spin$_3$-principal bundle given by the spin structure. Now the algebraic number of real curves, counted with respect to their spinor orientation, turns out to be independent of the choice of the configuration of points and this is my invariant.

Question: Is it possible to compute this invariant with the help of tropical algebraic geometry?

(contributed by Jean-Yves Welschinger)

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