Fibration with fibre $\mathbb{C}P^\infty$

Question

I have a fibration $F\to E\to B$ where $B$ is a nice space (a compact manifold) and the fibre is $\mathbb{C}P^\infty$, i. e. an Eilenberg–MacLane space $K(\mathbb{Z},2)$. Are there good criteria to check if this fibration is fibre homotopy equivalent to the trivial bundle? (I found here something for $K(G,1)$-fibrations)