Here is a Kahler example. Consider a hyper-elliptic curve $C$ with involution $\sigma$. Take $C\times S^1$ and quotient $C\times S^1$ by $\mathbb Z_2$ that rotates $S^1$ by a half-turn and acts by $\sigma$ on $C$. Now, to get a Kahler structure on $M\times M$ take $C\times C\times E$, where $E$ is an elliptic curve and quotient by an obvious action of $(\mathbb Z_2)^2$.

Here is a Kahler example. Consider a hyper-elliptic curve $C$ of positive genus with involution $\sigma$. Take $C\times S^1$ and quotient $C\times S^1$ by $\mathbb Z_2$ that rotates $S^1$ by a half-turn and acts by $\sigma$ on $C$. Now, to get a Kahler structure on $M\times M$ take $C\times C\times E$, where $E$ is an elliptic curve and quotient by an obvious action of $(\mathbb Z_2)^2$.