The following are well-known facts on the Serre spectral sequence
For a fibration $F \rightarrow E \rightarrow B$ we have the Serre spectral sequence (in cohomology with a coefficients in a field $k$) with $E_2$-term
$$E_2^{p,q} = H^p(B; \mathcal{H}^q(F)) \Rightarrow H^{p+q}(E) $$
If $B$ is simply connected, (or $\pi_1(B)$ acts trivially on the cohomology of the fiber) we have that
$$E_2^{p,q} = H^p(B) \otimes H^q(F) $$
Moreover, if the spectral sequence degenerates at the $E_2$-term ($d_r = 0$ for $r \geq 2$),
then $H^*(B) \otimes H^*(F) \cong H^*(E)$.
My question is the following,
Suppose that for a fibration $F \rightarrow E \rightarrow B$ we know that $H^*(E) \cong H^*(B) \otimes H^*(F)$, does it follows that in the spectral sequence $E_2^{p,q} = H^p(B) \otimes H^q(F) $ ? and also that $d_r = 0$ for $r \geq 2$ ?
I assume that it is not true and a counterxample should involve a non-trivial local coefficient system, but I do not know many "computable" examples where the base space is non-simply connected.