A bi-Lagrangian structure is a quadruple $(M,\omega,F,G)$, where $(M,\omega)$ is a symplectic manifold and $F, G$ are complementary foliations of $M$ with Lagrangian leaves. In his paper from 1993, Tabachnikov gave several interesting examples of bi-Lagrangian structures, including a certain bi-Lagrangian structure on a standard $2n$-dimensional symplectic space foliated by $F$ and $G$ into affine tangent spaces of two generic Lagrangian submanifolds $L,K$ respectively. He encouraged his readers to consider the following interesting question: for which Lagrangian submanifold-germs $L,K$ the corresponding bi-Lagrangian structure of the above kind is flat with respect to its canonical (bi-Lagrangian) connection? In this talk we give an answer to this question for $n=1$ and report on our progress regarding the general case. Joint work with W. Domitrz.