Let $M$ be a (compact if this somehow matters) smooth manifold with a involution $\theta: M \to M$, i.e. a smooth map such that $\theta^2 = \text{Id}$. A typical example is $M$ a $d$-torus and $\theta(x) = -x$.
Let $E \to M$ be a complex vector bundle over $M$, since the fibers a simply connected the long exact sequence of homotopy groups yields that $\pi_1(E) \to \pi_1(M)$ is onto and therefore there is a map $\Theta: E \to E$ over $\theta$.
I am interested in the case where $\Theta$ is a real vector bundle map which is complex anti-linear and $\Theta^2 = \pm\text{Id}$, is there any topological obstruction to the existence of such a morphism? If there is such morphism how many of them are?
Since the base map is not necessarily the identity this wont imply that $E \cong \bar{E}$, its conjugate bundle, and neither $c_1(E) = c_i(\bar{E}) = (-1)^i c_i(E)$, hence $c_i(E) = 0$ when $i$ is odd.
Is it possible to think this as a reduction of $G$-structure?