In the paper I am currently reading, there is the following setting. We have a complex oriented cohomology theory $E$, a complex orientation $x \in E^2(\mathbb{C}P^{\infty})$, the inclusion $i \colon \mathbb{R}P^{n} \to \mathbb{C}P^{\infty}$ and the induced element $x_{\mathbb{R}} := i^*(x) \in E^2(\mathbb{R}P^{n})$ for any $n$.
Now there is an element $x_{\mathbb{R}} \otimes 1 \in E^2(\mathbb{R}P^n \times \mathbb{R}P^m)$, but I dont know really how to make sense of this. If $E$ was the ordinary cohomology, one could just use the Künneth-formula, but for generalised cohomology, what can we say about the cohomology of this product, and if it really involves a tensor product?