For a field $k$ let $L_k$ denote the affine scheme $\mathbb{A}^1_k = \mathrm{Spec}(k[x])$.
The inclusion $\Bbb R[x] \hookrightarrow \Bbb C[x]$ induces a scheme morphism $f : L_\mathbb{C}\rightarrow L_\mathbb{R}$, which should be surjective on points (by e.g. the going-up theorem).
If I'm not mistaken, the underlying continuous map of $f$ sends points as follows: a max ideal $(x-a)$ for $a\in \mathbb{C}$ is sent to the minimal polynomial $(x-a)(x-\bar{a})$ of $x-a$ over $\mathbb{R}$. This is because the minimal polynomial generates $\mathbb{R}[x] \cap (x-a)$. Note both $a,\bar{a}$ are sent to the same point. Furthermore, this clearly sends non-conjugate $a,b\in\mathbb{C}$ to different points (different minimal polynomials).
I was wondering if it makes sense to say there should be a group scheme $\mathbb{Z}/2\mathbb{Z}$ acting on the scheme $L_\mathbb{C}$ by "complex conjugation", and the quotient gives the scheme $L_\mathbb{R}$? How would this be formalized in detail?