Let $(X,\omega)$ be a smooth $2n$-dimensional symplectic manifold. Suppose there is a free Hamiltonian $S^{1}$ action on $X$ with moment map $\mu:X\rightarrow\mathbb{R}$. Let $X^{+}=\{\mu\geq 0\}$ and $X^{-}=\{\mu\leq 0\}$. Given a smooth $(n-1)$-dimensional manifold $M$, a map $f:X\rightarrow\mathbb{R}\times M$ is said to be a stitched Lagrangian fibration if there is a continuous $S^{1}$ invariant function $G:X\rightarrow M$, such that the following holds:

• (i)

  Let $G^{\pm}=G|_{X^{\pm}}$. Then $G^{+}$ and $G^{-}$ are restrictions of $C^{\infty}$ maps on $X$;

• (ii)

  $f$ can be written as $f=(\mu,G)$ and $f$ restricted to $X^{\pm}$ is a proper submersion with connected Lagrangian fibres.

We call $Z=\mu^{-1}(0)$ the seam and $\Gamma=f(Z)\subseteq\{0\}\times M$ the wall. We denote $f^{\pm}=f|_{X^{\pm}}$.