Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are nice spaces (say smooth manifolds). Then $E$ has the form of a twisted product

(i) $E \cong \widetilde{B} \times_{\pi_{1}} F$,

where $\widetilde{B}$ is the universal cover of $B$, $\pi_{1}$ is the fundamental group of $B$ and $F$ carries a $\pi_{1}$-action.

Now, how nice can we assume this $\pi_1$-action to be? In particular, is it very restricting to only look at flat fiber bundles of the form

(ii) $E \cong \widetilde{B} \times_{\pi_{1}} |L|$,

where $L$ is a simplicial $\pi_{1}$-complex? What, if we further assume the action on $L$ to be regular?

I can't think of a flat bundle that is not of form (ii). Do you know counterexamples that clarify what (ii) cannot describe? (possibly with relaxed conditions on $E$, $F$, $B$.) And are there theorems that specify exact conditions on a flat bundle to be of form (ii)?

I am particularly interested in the case $B$, $F$ compact, $\pi_{1}$ infinite.