For Calabi-Yau variety $X$ which is a complete intersection $$ f_1=f_2=\ldots=f_r=0 $$ in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the following way. One can consider $X$ as zeroes of a section of vector bundle $$ {\mathcal O}_{{\mathbb P }^n}(d_1)\oplus\ldots\oplus{\mathcal O}_{{\mathbb P }^n}(d_r) $$ with $d_i=\mathrm{deg}\,(f_i)$. This section defines a function $W$ on the (total space of) dual bundle $$ {\mathcal X}={\mathrm{Total\ space\ of\ }}{\mathcal O}_{{\mathbb P }^n}(-d_1)\oplus\ldots\oplus{\mathcal O}_{{\mathbb P }^n}(-d_r). $$ One can check that $$ W=y_1f_1+\ldots+y_rf_r $$ for $y_i$ defined in some way. One has Landau-Ginzburg model: $$ W:{\mathcal X} \to {\mathbb C}. $$

Question: Is it possible to construct another Landau-Ginsburg model $$ W':{\mathcal X}'\to {\mathbb C} $$ such that (1) ${\mathcal X}'$ is a toric variety (as well as ${\mathcal X}$) and (2) one can not get ${\mathcal X}'$ from $\mathcal X$ by a sequence of toric blow-ups and contractions.

Comment 1. (a) It is not necessary that $\mathcal X'$ is also a total space of some vetor bundle, (b) it is not necessary that ${\mathrm{dim}}\,{\mathcal X}={\mathrm{dim}}\,{\mathcal X}'$.

Comment 2. In other words, I am looking for a toric variety $\mathcal X'$ such that the support of its fan (it should be a cone and I would like to have this cone simplicial) would be different from support of the fan for $\mathcal X$.