Given a compact simple integral affine $3$-manifold with singularities $(B,\Delta,\mathscr{A})$, all of whose negative vertices are straight (i.e. locally isomorphic to Example 3.12), there is a localized thickening $(\Delta_{\blacklozenge},\{D_{p^{-}}\}_{p^{-}\in\mathcal{N}})$ of $\Delta$ and a smooth, compact symplectic $6$-manifold $(X,\omega)$ together with a piecewise smooth Lagrangian fibration $f:X\rightarrow B$ such that

• (i)

  $f$ is smooth except along $\bigcup_{p^{-}\in\mathcal{N}}\,f^{-1}(D_{p^{-}})$;

• (ii)

  the discriminant locus of $f$ is $\Delta_{\blacklozenge}$;

• (iii)

  there is a commuting diagram

  $$\begin{CD}X(B_{\blacklozenge},\mathscr{A}_{\blacklozenge})@>{\Psi}>{}>X\\ @V{f_{0}}V{}V@V{}V{f}V\\ B_{\blacklozenge}@>{\iota}>{}>B\end{CD}$$

  where $\psi$ is a symplectomorphism and $\iota$ the inclusion;

• (iv)

  over a neighborhood of a positive vertex of $\Delta_{\blacklozenge}$ the fibration is positive, over a neighborhood of a point on an edge the fibration is generic-singular, over a neighborhood of $D_{p^{-}}$ the fibration is Lagrangian negative.