  We introduce a general technique to construct Lagrangian torus fibrations in
degenerations of K\"ahler manifolds. We show that such torus fibrations
naturally occur at the boundary of the A'Campo space. This space extends a
degeneration over a punctured disc to a hybrid space involving its real
oriented blowup and the dual complex of the special fiber; equipped with an
exact modification of a given K\"ahler form which becomes fiberwise symplectic
at the boundary. Therefore, it allows to move Lagrangian tori from the "radius
zero" fibers at the boundary to the nearby fibers of the original generation
using the symplectic connection. As a consequence, for any maximal Calabi-Yau
degeneration we prove that a generic region of each nearby fiber admits a
Lagrangian torus fibration with asymptotic properties expected from the
(conjectural) Strominger--Yau--Zaslow fibration.
