Let $G$ be a finite group and let $V$ be a finite dimensional real representation of $G$. I'm thinking of $V$ as a module over $\mathbb{R}[G]$.
Here are two ways to make a $\mathbb{C}[G]$-module out of $V$.
Take the tensor product of $\mathbb{C}[G]$ with $V$ over $\mathbb{R}[G]$ (where $\mathbb{C}[G]$ is viewed as a $\mathbb{C}[G]$-$\mathbb{R}[G]$-bimodule.
Take the tensor product of $\mathbb{C}$ with $V$ over $\mathbb{R}$, and equip it with an action of $g$ given by $g\cdot(z\otimes v)=z\otimes(gv)$ (and show that this rule really does extend to a well-defined additive action of $\mathbb{C}[G]$).
I think that I was able to show that both representations are isomorphic. I showed that the module in (2) satisfies the universal property required from the tensor product in (1).
Am I correct? Are these two modules really isomorphic?
Yes. This also follows by composing the canonical isomorphisms
$$\mathbf{C} \otimes_\mathbf{R} V \leftarrow \mathbf{C} \otimes_\mathbf{R} \mathbf{R}[G] \otimes_{\mathbf{R}[G]} V \rightarrow \mathbf{C}[G] \otimes_{\mathbf{R}[G]} V.$$