The Klein quartic $\chi$ is given by the set of solutions to the homogeneous equation $$x^3y + y^3z + z^3x=0$$ in $\mathbb{C}P^2$. This has 168 orientation preserving automorphisms, including copies of the 12 element tetrahedral symmetry group.

Is there a nice way to take the points of $\chi$ in $\mathbb{C}P^2$, map them to $\mathbb{R}^3$ (preserving a tetrahedral symmetry group), and produce an embedded genus three surface?

There are a number of models of the Klein quartic in $\mathbb{R}^3$ out there, for example by Joe Christy and Greg Egan (see this webpage by John Baez) and this version by Carlo Sequin. As far as we (Saul Schleimer and I) can tell, these are all "topological" models, not mapping from $\chi\subset \mathbb{C}P^2$ directly in some sensible way.