  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.
