No, I don't want to consider complete intersection, I just want to consider A-model where I compactify $(\C^*)^n$ somehow.

For example, $X_A = (\C^*)^3 \cup D_{1,1,1}$, and $W_A = x+y+z$. And the mirror is $X_B = \C^3$, with $W_B = xyz$.

Suppose I take products of these type. I get HMS still.

Then, I consider some torus action on the B-side, and fibration to dual torus on the A-side. We have $$ Coh([X_B / T], W) = Coh(X_B,W_B)^T \cong Fuk(X_A, W_A)^T \cong Fuk(\wt X_A, \wt W_A) $$

Now, here is the interesting thing: if we take GIT quotient, that corresponds to localization by the unstable loci.