Clearly, given any CM field $E$, it is known that every subfields of $E$ is either totally real or totally imaginary. Moreover, we can consider $F$ the $\bf{maximum ~totally~ real~ subfield}$ of $E$.
So, I wonder if there is $\textit{only one}$ such $F$ in $E$ or under some conditions this fact is true.
The compositum of any two totally real fields is totally real, so in fact for any number field $E$ there is a unique maximal totally real subfield $F$, which is the compositum of all the totally real subfields of $E$.