This was originally posted on MSE, and since it didn't receive much attention, I'll try here. Let me know if this is not the appropriate place.

Given a fiber bundle $F \to E \to B$ over a paracompact base $B$, assume its cohomology satisfies all the required properties for the Leray-Hirsch theorem to hold. This tells us that, as groups, $$ H^n (E,R) \cong \bigoplus_{p+q=n} H^p (F,R) \otimes H^q (B, R). $$

But this does NOT tell us that, as rings, we have a ring isomorphism $$ H^* (E,R) \cong H^* (F,R) \otimes H^* (B,R). $$

I'm wondering when this stronger isomorphism actually does hold. What extra structure on the fiber bundle is sufficient, without it being a trivial bundle?

I know of the classic counterexample with the bundle $\mathbb{C}P^3$ over $S^4$ with fiber $S^2$. I know that principle bundles can have the module isomorphism of Leray-Hirsch upgraded to the ring isomorphism. What else do we know? Any references on this subject would be highly appreciated.

**EDIT:** How about the following rather strong condition on our fiber bundle. Say both the integral cohomology of $F$ and $B$ lie in even degrees only, and that the bundle is torsion-free (one of the two cohomology rings are free and finitely generated, I believe is the criteria). Then, when applying the Serre spectral sequence, we would immediately get that it collapses for degree reasons. I only have a basic understanding of the SSS, but would this be enough to show that $H^* (E, \mathbb{Z})$ splits as a tensor products of the two other cohomology rings?