The $(d-1)$-torus fibration $W(\lambda,\theta)$ (more naturally, a $\mathbb{T}^{d-1}$-torsor) will depend on additional phase multi-parameter $\theta:=\{\theta_{v}\},v\in\operatorname{vert}(S)$, where all $\theta_{v}$ have values in $\mathbb{R}/\mathbb{Z}$. First, we form (trivial) affine torus bundles over the affine open sets by identifying the fibers with the affine tori

	$$\displaystyle\mathbb{T}_{v}(\theta_{v}):=\{n\in\mathbb{T}\ :\ \langle v,n\rangle+\theta_{v}\equiv 0\mod\mathbb{Z}\}\quad\text{ over }U_{v},$$
	$$\displaystyle\mathbb{T}/w:=(\mathbb{R}^{d}/w)/(\mathbb{Z}^{d}/w)\quad\text{ over }V_{w}.$$

The gluing maps are independent of a base point in an overlap $U_{v}\cap V_{w}$ and given there by the natural projection $\mathbb{T}_{v}(\theta_{v})\to\mathbb{T}/w$. This defines the torsor $\pi:W(\lambda,\theta)\to{\Sigma\backslash D}$.

The topology of the total space of the torsor is determined by the combinatorics of $\Sigma$, i.e., independent of $\lambda$ and $\theta$ as long as $\lambda$ is in the right secondary cone $\operatorname{SC}(S)$. In particular, all $W(\lambda,\theta)$ are diffeomorphic to each other, though not canonically.

Since the linear parts of the transition maps are the same for the base and for the fibers, the tangent space at any point in $W(\lambda,\theta)$ splits canonically as

$$T_{W(\lambda,\theta)}\cong T_{\Sigma\backslash D}\oplus T_{\Sigma\backslash D}.$$

This allows to define a canonical (integrable) almost complex structure on $W(\lambda,\theta)$ as

$$J_{w}=\left(\begin{array}[]{cc}0&\mathbbm{1}\\ -\mathbbm{1}&0\\ \end{array}\right).$$

Given a Riemannian metric $g_{ij}$ on $Y$, one can define the pullback metric $\pi^{*}(g_{ij})$ on $W(\lambda,\theta)$ which is, in fact, Kähler. If, in addition, $g_{ij}$ satisfy the real Monge-Ampère equation, the induced metric on $W(\lambda,\theta)$ is Ricci-flat.

There is another slightly different description of the torus bundle over a Kähler affine manifold $Y$ (cf. [KS01]), which is useful when considering limiting behavior of Calabi-Yau degenerations. We define a torus fibration $W_{\epsilon}(\lambda)$ as the quotient of the total space of the tangent bundle $T({{\Sigma\backslash D}})$ by the integral lattice spanned by $\left\{\epsilon\frac{\partial}{\partial y_{i}}\right\}$, where $y_{i}$ are the affine coordinates. This torus bundle carries canonical complex structure and the pullback metric which comes from the splitting $T_{W_{\epsilon}(\lambda)}\cong T_{\Sigma\backslash D}\oplus T_{\Sigma\backslash D}$ as before.

But in order to make a connection with the previous picture and with the geometry of toric hypersurfaces we need to twist the complex structure on $W_{\epsilon}(\lambda)$ by the element of $H^{1}({\Sigma\backslash D},\mathbb{T}^{d-1})$ associated with the phase parameters $\theta_{v}$. The resulting Kähler manifold $W_{\epsilon}(\lambda,\theta)$ can be canonically identified with $W(\epsilon^{-1}\lambda,\theta)$ constructed by the first method starting with the $\epsilon$-rescaled Kähler affine structure.