Large volume limit fibrations over fanifolds

We lift the stratified torus fibration over a fanifold constructed by Gammage--Shende to the associated Weinstein manifold-with-boundary, which is homotopic to a filtered stratified integrable system with noncompact fibers. When the fanifold admits a dual stratified space in a suitable sense, we give a stratified fibration over it completing SYZ picture. For the fanifold associated with a very affine hypersurface, we realize the latter fibration as a restriction of SYZ fibrations over the tropical hypersurface proposed by Abouzaid--Auroux--Katzarkov.