Let $f\colon E\to B$ locally trivial bundle of 'nice' topological spaces (say finite CW-complexes) with fiber $F$. Assume also that the base $B$ is simply connected.

Assume that either the cohomology spectral sequence (with coefficients in a field) degenerates in the second term $H^*(B)\otimes H^*(F)$ or that the push-forward in the derived category of the constant sheaf is isomorphic (in derived category) to the direct sum of its cohomology sheaves (I think that the second condition implies the first one, but I am not sure about the converse).

Is it true that there exists an isomoprhism of graded algebras $$H^*(E)\simeq H^*(B)\otimes H^*(F)$$ which is compatible with the pull-back $f^*\colon H^*(B)\to H^*(E)$ and with restriction to fiber $H^*(E)\to H^*(F)$?