I am looking for a somewhat canonical reference for the following fact. Let $L$ be $2$-dimensional real bundle over $\mathbb{CP}^{1}$ with smooth non-degenerate bilinear form $\omega$ on fibres. Then we may endow $L$ with the structure of a holomorphic vector bundle $J \in End(L)$ in a way that is compatible with $\omega$, i.e. $\omega(J \cdot, \cdot)$ is an inner product on the fibres.
Note the existence of a compatible almost complex structure is guaranteed by proposition 2.63 of "Introduction to symplectic topology" by Mcduff and Salamon.