Real line subbundles of real bundles on curves

For a stable real bundle $E$ of rank $2$ and degree $1$ on a real genus $2$ curve, we describe the action of the real structure of the curve on the set of $4$ maximal line subbundles of degree $0$ of $E$. This describes the Galois action on the set of lines through a real point in the moduli space of such bundles, and is a real algebraic extension of classical work of Newstead. Our proof is an application of techniques of Atiyah from the 1950’s. We prove also results on real line subbundles in higher genus using work of Lange-Narasimhan.