  We construct a family of Lagrangian submanifolds in the Landau--Ginzburg
mirror to the projective plane equipped with a binodal cubic curve as
anticanonical divisor. These objects correspond under mirror symmetry to the
powers of the twisting sheaf O(1), and hence their Floer cohomology groups form
an algebra isomorphic to the homogeneous coordinate ring. An interesting
feature is the presence of a singular torus fibration on the mirror, of which
the Lagrangians are sections. The algebra structure on the Floer cohomology is
computed by counting sections of Lefschetz fibrations. Our results agree with
the tropical analog proposed by Abouzaid--Gross--Siebert. An extension to
mirrors of the complements of components of the anticanonical divisor is
discussed.
