 Clean-circle resolution of a Lagrangian surface immersion

Perturb a clean self-intersection circle of a Lagrangian immersion by a Morse function on $S^1$. Its minimum and maximum give two transverse double points. Resolving both by Lagrangian surgery adds two one-handles and changes the Euler characteristic by $-4$. One neck may be glued with the orientation-reversing choice, so the resolved surface is nonorientable.