If $X$ is a K3 surface, it carries a unique (up to scaling with $\mathbb{C}^*$) holomorphic 2-form $\sigma$, determined by the complex structure. Let $\Lambda=3U\oplus-2E_8$ be the K3-lattice. The period domain of a K3 surface is given by $$ \Omega=\{[\sigma]\in \mathbb{P}(\Lambda\otimes\mathbb{C})\colon \sigma\cdot\sigma=0, \sigma\cdot\overline{\sigma}>0\}. $$ A marked K3 surface is a pair $(X,\phi)$, with $X$ K3 and where $\phi$ is an isometry $\phi:H^2(X,\mathbb{Z})\to \Lambda$. Let $M_1$ be the set of equivalence classes of Marked K3 surfaces, and define the Period map $\tau:M_1\to \Omega$ by $\tau([X,\phi])=[\phi(\sigma_X)]$.
Now suppose $Y$ is another K3 surface, and consider the product of K3 surfaces $X\times Y$. I want to define the period map for this product. By Künneth's theorem, the natural map $$H^2(X,\mathbb{Z})\oplus H^2(Y,\mathbb{Z})\to H^2(X\times Y,\mathbb{Z})$$ is an isomorphism, so this endows $H^2(X\times Y,\mathbb{Z})$ with the natural lattice structure, isomorphic to $\Lambda\oplus\Lambda$. Let $\phi:H^2(X\times Y,\mathbb{Z})\to \Lambda\oplus\Lambda$. The product $X\times Y$ carries two nowhere vanishing holomorphic 2-forms: $\sigma_1=\pi^*_X\sigma_X$ and $\sigma_2=\pi^*_Y\sigma_Y$. The new period domain would be $$ \Omega'=\{[(\sigma_1,\sigma_2)]\in\mathbb{P}((\Lambda\oplus\Lambda)\otimes\mathbb{C})\colon \sigma_1\cdot\sigma_1=0,\sigma_1\cdot\sigma_2=0,\sigma_2\cdot\sigma_2=0,\sigma_1\cdot\overline{\sigma_2}=0, \sigma_1\cdot\overline{\sigma_1}>0,\sigma_2\cdot\overline{\sigma_2}>0\}, $$ and I would like to define $\tau':M_1'\to \Omega'$ by $\tau'([X\times Y, \phi])=[(\phi(\sigma_1),\phi(\sigma_2)]$. However, this is not really well-defined. Is there a better way to define the period map for the product of K3 surfaces?
Edit: At first, I thought that $M_1'=M_1\times M_1, \Omega'=\Omega\times \Omega$ and $\tau'=\tau\times\tau$. However, this does not seem to work dimensionwise, so I am interested in any ideas!