Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question concerns the condition, let's call it (*), that $\tau$ is in the center of $G$. In other words, the condition says that $\sigma(\overline{\alpha}) = \overline{\sigma(\alpha)}$ whenever $\sigma \in G$ and $\alpha \in K$.

Some observations:

-(*) is satisfied if $K \subseteq \mathbb{R}$, since $\tau$ is trivial then, and

-the condition (*) is very restrictive on the structure of $G$ if $\tau$ is not trivial.

I feel like I have seen this condition in literature before, but I can't remember where, hence this question:

Can you provide any references in the literature to this condition (such as theorems where it is a hypothesis or conclusion)?

See also this slightly related post: Centraliser of the complex conjugation in the absolute Galois group

One may also ask whether (*) holds for every embedding $K \hookrightarrow \mathbb{C}$, so references to this condition would also be appreciated.