  Let $E$ be a smooth bundle with fiber an $n$-dimensional real projective
space $\mathbb{R}P^n$. We show that, if every fiber carries a positively curved
pointwise strongly $1/4$-pinched Riemannian metric that varies continuously
with respect to its base point, then the structure group of the bundle reduces
to the isometry group of the standard round metric on $\mathbb{R}P^n$.
