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 $$ H^n (E,R) \cong \bigoplus_{p+q=n} H^p (F,R) \otimes H^q (B, R), $$ where the isomorphism is in terms of $R$-modules.
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 upgraded to the ring isomorphism. What else do we know? Any references on this subject would be highly appreciated.