Let $X$ be a scheme defined over $\mathbb{C}$ with an involution $\sigma$. How to get a $X_{\mathbb{R}}$ scheme defined over $\mathbb{R}$ such that $X_{\mathbb{R}}\times_\mathbb{R} \mathbb{C} = X$ ? How are the real algebraic bundle on $X_{\mathbb{R}}$ and complex bundle on $X$ related?

The paper by Atiyah on Ktheory and reality Quart. J. Math. Oxford Ser. (2) 17 1966 367386. MR0206940 discusses the topological analogue of this question, showing how to relate complex vector bundles over a space with vector bundles over the fixed points of an involution. The idea is to define a sort of intermediate Kgroup KR of vector bundles with involution, and compare it to the two other Kgroups. 

