This is in Moonen's textbook on Abelian Varieties (page 221). My confusion I believe is that I do not understand how to think about $\mathbb{R}$-varieties too well (having only ever worked over $\mathbb{C}$ or $\mathbb{F}_q$ all my life so far).
It is claimed that the curve $X^2+Y^2+Z^2=0$ in $\mathbb{P}^2$ over $\mathbb{R}$ has no line bundles of odd degree. However, this curve has no $\mathbb{R}$-points and so a subset of this projective space, it is empty. So there shouldn't be any line bundles whatsoever on it. What am I missing here?