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We associate real and regular algebraic–geometric data to each multi-line soliton solution of Kadomtsev–Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non-negative part of real Grassmannians $$Gr^{\mathrm{TNN}}(k,n)$$GrTNN(k,n). In [3] we were able to construct real algebraic–geometric data for soliton data in the main cell $$Gr^{\mathrm{TP}} (k,n)$$GrTP(k,n) only. Here we do not just extend that construction to all points in $$Gr^{\mathrm{TNN}}(k,n)$$GrTNN(k,n), but we also considerably simplify it, since both the reducible rational $$\texttt {M}$$M-curve $$\Gamma $$Γ and the real regular KP divisor on $$\Gamma $$Γ are directly related to the parametrization of positroid cells in $$Gr^{\mathrm{TNN}}(k,n)$$GrTNN(k,n) via the Le-networks introduced in [63]. In particular, the direct relation of our construction to the Le-networks guarantees that the genus of the underlying smooth $$\texttt {M}$$M-curve is minimal and it coincides with the dimension of the positroid cell in $$Gr^{\mathrm{TNN}}(k,n)$$GrTNN(k,n) to which the soliton data belong to. Finally, we apply our construction to soliton data in $$Gr^{\mathrm{TP}}(2,4)$$GrTP(2,4) and we compare it with that in [3].

Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons