This is more of a comment than an answer, because I think this is a complicated story and it may just be better to give the relevant references. In short, this has been solved partially by (Kontsevich and Soibelman) in
and then completely by (Gross and Siebert) when the singular locus of the fibration has real codimension 2. These works are mostly concerned with the problem of how we can recover the mirror given a singular affine structure with mild singularities. Gross has a big book on his website which surveys this body of work. He also recently released a shorter 65 page survey
which also touches on this topic extensively, though I haven't had a chance to look at it closely yet. It seems important to note that many Lagrangian fibrations in nature have codimension 1 singularities. Of course, nobody knows how to write down (special) Lagrangian fibrations with the desired properties on Calabi-Yau threefolds. Gross and Siebert seem to work around these hard analytical issues using toric degeneration and staying in the realm of algebraic geometry.
One could also consult work by Auroux e.g. http://arxiv.org/abs/0706.3207 for concrete examples where one works with actual Lagrangian fibrations and shows how wall-crossing is natural from the point of view of symplectic geometryconsiderations. Note that Auroux is concerned with a slightly different setup then than the OP (that of mirror symmetry in the complement of an anticanonical divisor). I think consulting those sources would be give a lot more information informative than any short answer.