Given a continuous group homomorphism $\rho : G\rightarrow \mathrm{GL}(k,\mathbb{R})$, by a $G$-structure (or $\rho$-structure if we want to be specific) on a given rank-$k$ real vector bundle $V\rightarrow X$, we mean a lift of $V$'s classifying map $X\rightarrow B\mathrm{GL}(k,\mathbb{R})$ (defined up to coherent homotopy) along $B\rho$ to a map $X\rightarrow BG$.
This gives rise to a principal $G$-bundle $E\rightarrow X$ and a $G$-equivariant map $E\rightarrow \mathrm{Frames}(V)$, lying over (the identity map of) $X$.
We say a $G$-structure $E\rightarrow X$ with $E\rightarrow \mathrm{Frames}(V)$ on a vector bundle $V\rightarrow X$ is "integrable" if, for every $x \in X$, there is an open neighborhood $U$ of $x$ in $X$, such that the restricted bundle $E|_U$ is isomorphic to the trivial $G$-bundle on $U$.
We say the structure is "integrable on stalks" if, for every $x \in X$, the stalk of $E$ at $x$ is isomorphic to that of the trivial principal $G$-bundle.
I had a few questions about these definitions:
Are "integrability" and "integrability on stalks" not equivalent?
Since $E\rightarrow X$ is a (continuous) principal $G$-bundle, shouldn't it always be "locally free", i.e. locally isomorphic to a trivial principal $G$-bundle? Why don't the above integrability conditions always hold?