Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, $c(z)=\frac{-1}{\bar{z}}$, and after all suppose we have a commutative diagram (I couldn't draw it here, imagine it yourself)

$\pi: E \rightarrow \bar{E} $

where $\pi$ is an anti-holomorphic (or anti-complex linear) involution covering $c$. In this situation, is it true that $c_1(E)$ should be even!