One standard way to compactify $\mathbb{R}^n$ is the "one-point" compactification to the sphere $S^n$. But what if I want to compactify it to a torus $T^n$ instead? What I have in mind is compactifying each coordinate direction to an $S^1$, and then the product $S_1 \times S_1 \times \dots \times S_1 = T^n$.
Is this a standard construction, or does it have a name? Any references to such a construction appearing in the literature would be appreciated.
It does not have a special name. It is just the more or less obvious fact that if you have a family of spaces $X_\alpha$ and a family $cX_\alpha$ of compactifications of $X_\alpha$, then $\prod_\alpha cX_\alpha$ is a compactification of $\prod_\alpha X_\alpha$.