Let $X$ and $Y$ be connected smooth manifolds. Let $L$ be a topological real line
bundle over $X\times Y$. Then we know that the isomorphism class of such a line bundle is determined by its first Stiefel-Whitney class $w_1(L)\in H^1(X\times Y,\mathbf{Z}/2\mathbf{Z})$.

I would like to have an example of a nontrivial line bundle $L$ (so $w_1(L)\neq 0$), such that

**(1)** $\forall y\in Y$ we have $w_1(L)|_{X\times\{y\}}=0\in H^1(X\times\{y\},\mathbf{Z}/2\mathbf{Z})$. 

**(2)**  $L$ is not the pullback of a line bundle over $Y$.

I don't see how to construct such a line bundle. May be there is some trick using Kunneth's formula.