  In this paper, we review or introduce several differential structures on
manifolds in the general setting of real and complex differential geometry, and
apply this study to Teichm\"uller theory. We focus on bi-Lagrangian i.e.
para-K\"ahler structures, which consist of a symplectic form and a pair of
transverse Lagrangian foliations. We prove in particular that the
complexification of a real-analytic K\"ahler manifold has a natural complex
bi-Lagrangian structure. We then specialize to moduli spaces of geometric
structures on closed surfaces, and show that old and new geometric features of
these spaces are formal consequences of the general theory. We also gain
clarity on several well-known results of Teichm\"uller theory by deriving them
from pure differential geometric machinery.
