It's pretty rare for a multiplicative cohomology theory $E$ to have a Kunneth isomorphism $E^\ast(X \times Y) \cong E^\ast(X) \otimes_{E^\ast(pt)} E^\ast(Y)$ for all spaces $X,Y$. Are there any examples of cohomology theories $E$ which have a Kunneth isomorphism just for powers, but not for all products of spaces? That is,

Question:Are there examples of multiplicative cohomology theories $E$ such that for all spaces $X$ and all $n \in \mathbb N$, the canonical map $$E^\ast(X) \otimes_{E^\ast(pt)} \dots \otimes_{E^\ast(pt)} E^\ast(X) \to E^\ast(X\times \dots \times X)$$ is an isomorphism (where the tensor product and cartesian product are taken $n$ times), and yet there are spaces $X,Y$ such that the canonical map $$E^\ast(X) \otimes_{E^\ast(pt)} E^\ast(Y) \to E^\ast(X \times Y)$$ is not an isomorphism?

I'd be happy to understand this question in slightly different contexts, such as assuming $X,Y$ to be finite, or taking them to be spectra or pointed spaces rather than spaces, or assuming that $E$ has more structure.

As alluded to at the start, the condition that $E$ doesn't have a Kunneth isomorphism in general doesn't rule out too many $E$'s -- I think just $H\mathbb F_p$, $K(n)$, $H \mathbb Q$, and things constructed from them.