Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$.
The complex conjugation $f$ is not holomorphic, but $C^{\infty}$. So we can look at $f^{*}E$ as a smooth real vector bundle. Since $f$ is an involution $E$ and $f^{*}E$ are isomorphic as smooth real bundles (this is most likely wrong). But somehow there should be more structure on $f^{*}E$. I mean $f$ is antiholomorphic, that should be useful somehow.
Is there a way to define a (canonical?) complex structure on $f^{*}E$, such that for any point $p$ we have an antilinear isomorphism $E_{p} \rightarrow f^{*}E_{p}=E_{\overline{p}}f^{*}E_{p}=E_{\overline{p}}$?
I think we don't get a linear iso, since the complex conjugation itself is only antilinear.
