A complex-analytic characterization of Lagrangian immersions in ${\mathbb{C}}^n$ with transverse double points

Given a compact smooth totally real immersed $n$-submanifold $M\subset {\mathbb{C}}^n$ with only finitely many transverse double points, it is known that if $M$ is Lagrangian with respect to some K{ä}hler form on ${\mathbb{C}}^n$, then it is rationally convex in ${\mathbb{C}}^n$ (Gayet, 2000), but the converse is not true (Mitrea, 2020). We show that $M$ is Lagrangian with respect to some K{ä}hler form on ${\mathbb{C}}^n$ if and only if $M$ is rationally convex {\em and} at each double point, the pair of transverse tangent planes to $M$ satisfies the following diagonalizability condition: there is a complex linear transformation on ${\mathbb{C}}^n$ that maps the pair to $({\mathbb{R}}^n,(D+i){\mathbb{R}}^n)$ for some real diagonal $n\times n$ matrix $D$.