Consider the inclusion $j$ of ${\mathbb{R}}$ as the real axis of ${\mathbb{C}}$. On ${\mathbb{C}}$ I have a real polynomial algebra ${\mathbb{R}}[x,\bar{x}]$, where $\bar{x}$ denotes conjugation. Restricting to real values for $x$, this algebra collapses to the usual real polynomial algebra ${\mathbb{R}}[x]$ on ${\mathbb{R}}$. (This is the reason why I consider real algebra ${\mathbb{R}}[x,\bar{x}]$ instead of complex algebra ${\mathbb{C}}[x,\bar{x}]$.

Now I'm trying to be generalize this to the inclusion of a real variety to its corresponding complex variety defined by the same equations. The first step to do this is to realize the above construction in terms of direct images and inverse images of sheaves. That means given the real polynomial algebra ${\mathbb{R}}[x]$, I wish to express ${\mathbb{R}}[x,\bar{x}]$ via some direct image or inverse image operations. I've tried taking the real part $\mathrm{Re}:{\mathbb{C}}\to{\mathbb{R}}$ too, but it doesn't work.

Any suggestions? Or am I trying to force something unnatural?