Let $P^2 \tilde \times \mathbb R^2$ be the $\mathbb Z_2$-quotient of $S^2 \times \mathbb R^2$, where the $\mathbb Z_2$ action on $S^2 \times \mathbb R^2$ is antipodal on $S^2$ and a reflection on $\mathbb R^2$. Similarly, $K^2 \tilde \times \mathbb R^2$ is the $\mathbb Z_2$-quotient of $T^2 \times \mathbb R^2$.

Can we find a Kahler surface $M$ so that $M$ is homeomorphic to $P^2 \tilde \times \mathbb R^2$ or $K^2 \tilde \times \mathbb R^2$?