Is there any relation between real and complex character functions of irreducible representations of compact lie groups?

Question

Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $\chi_U^\mathbb{R}:G\to\mathbb{R}$ as $\chi_U^\mathbb{R}(g)=\operatorname{Tr}(l_g)$. If $V$ is a complex $G$-module one can define its character as $\chi_V:G\to\mathbb{C}$ as $\chi_V(g)=\operatorname{Tr}(l_g)$. Also, one can also define a extension map $e_+(U)=\mathbb{C}\otimes_\mathbb{R} U$ that maps a real $G$-module into a complex $G$-module. Is there any relation between $\chi_U^\mathbb{R}$ and $\chi_{e_+(U)}$ for irreducible $U$?