Let $X$ be a double cover of $\mathbb{P}^n \times \mathbb{P}^n$ branching along bi-degree $(d,e)$ hypersurface. Is it possible to write $X$ as a complete intersection of hypersurfaces in some weighted projective space (for certain $d,e$)?

This is motivated by the fact that a double cover of $\mathbb{P}^n$ branching along degree $2d$ hypersurface can be written as a hypersurface in $\mathbb{P}^n_{(1^{n+1},d)}$.