Composite of two fields contain a given quadratic extension, but each individual doesn't.

Question

In fact, this question could be asked for arbitrary field extension. However, for simplicity I only ask the question for local field of characteristic 0. Let $E/F$ be a quadratic extension of padic field (we may assume p≠2 also for simplicity). Can we classify all the possible cases: $E_1/F$ and $E_2/F$ are finite extensions of F, both of which are disjoint from E, but the composite $E_1E_2$ (we may also assume $E_1,E_2$ being disjoint from each other so that $E_1E_2\cong E_1\otimes_{F}E_2$) in $\overline{F}$ contains $E$. The most famous example is given by the biquadratic extensions $K/F$ such that $E_1,E_2,E$ are three intermediate quadratic extensions of $F$. Is there any example beyond this? Can we give a description when this happens?