For $V \rightarrow X$ a rank-$(n+1)$ real vector bundle, let $Z_V \subset V$ denote (the image of) its zero section. Let $E_V = (V\setminus Z_V)\,/\,(\mathbb{R}\setminus\{0\})$ be the quotient of $V$ without its zero section, by the action of the nonzero reals; now we get $E_V \rightarrow X$ a fiber bundle with fiber $\mathbb{R}\mathbb{P}^n$.
I was wondering if there is any description of the obstruction to existence of a global section of $E_V\rightarrow X$ as a characteristic class of $V\rightarrow X$? If not in general, perhaps in the case where $X$ is a compact smooth manifold, and $V\rightarrow X$ is the tangent bundle?
Here I'm thinking analogously to how if $X$ is a compact smooth manifold, then the mod-2 Euler characteristic of $TX\rightarrow X$ being nonzero tells us any (continuous) vector field on $X$ must have a zero. On the other hand, the situation described above allows for vector fields "up to a sign".