Let $G_1$ and $G_2$ be compact Lie groups. We know that each finite-dimensional complex irreducible representation of $G_1\times G_2$ is the tensor product of an irreducible representation of $G_1$ and an irreducible representation of $G_2$.

But for real representation of $G_1\times G_2$, **what is the general form of an irreducible real representation of the product group?**

I don't think tensor product of a real irreducible representation of $G_1$ and one of $G_2$ works.

For example, if $G_1=\mathbb{Z}/p\mathbb{Z}$, $G_2=\mathbb{R}$, let $\rho_1: \mathbb{Z}/p\mathbb{Z}\longrightarrow O(2)$ send $m$ to the matrix

$\cos \frac{2\pi m}{p}, -\sin \frac{2\pi m}{p}$

$\sin \frac{2\pi m}{p}, \cos \frac{2\pi m}{p}$

and let $\rho_2: \mathbb{R}\longrightarrow O(2)$ send $t$ to the matrix

$ \cos at, -\sin at$

$\sin at, \cos at$

then the representation $G_1\times G_2\longrightarrow O(2)$ sending $(m, t)$ to

$\cos (at+\frac{2\pi m}{p}), -\sin (at+\frac{2\pi m}{p})$

$\sin (at+\frac{2\pi m}{p}), \cos (at+\frac{2\pi m}{p})$

is a real irreducible representation of $G_1\times G_2$ with representation space $\mathbb{R}^2$. It is definitely NOT $\rho_1\otimes \rho_2$, whose representation space is $\mathbb{R}^2\otimes\mathbb{R}^2=\mathbb{R}^4$. It's NOT the tensor product of any real irreducible representation $G_1$ and one of $G_2$ either.

I'm curious for any compact Lie group $G_1$ and $G_2$, whether there is a general form of an (irreducible) real representation of $G_1\times G_2$. So far I haven't found any reference on this.