Maximal Unramified subextension of Compositum

Question

Let $L/K$ and $M/K$ be two finite extensions of henselian fields. Let $T$ and $U$ be the maximal unramified subextensions of $L/K$ and $M/K$, respectively. Is it true that the maximal unramified subextension of $LM/K$ is $TU$?

Answer 1

The unique unramified quadratic extension of $\Bbb Q_2$ is $\Bbb Q_2(\sqrt{5})$. Can you find two ramified quadratic extensions of $\Bbb Q_2$ whose compositum contain $\Bbb Q_2(\sqrt5)$?