What is the $\mathbb{R}$-scheme $\operatorname{Spec} \mathbb{C}$?

Question

I want to study $\operatorname{Spec} \mathbb{C}$ in the category of $\mathbb{R}$-schemes. I know that the structural morphism is derived from the ring morphism $\mathbb{R} \to \mathbb{C}$, which must be the inclusion. I also know that, geometrically, both $\operatorname{Spec} \mathbb{C}$ and $\operatorname{Spec} \mathbb{R}$ are just a single point, but I think that $\operatorname{Spec} \mathbb{C}$ as a $\mathbb{R}$-scheme is obtained after having two points glued together. How can I show this?