To Reside
To Stay, To Remain
する verb
If 在留 is residence, then the verb version of that would be to reside.
Like 在留, this is also a more formal term. You might encounter it at the immigration office, but you probably won't hear it in everyday conversation.
The reading comes from 在留 so you should be able to read this one already!
外国人が日本に在留する時に必要になるものは何ですか？
What do foreigners need when they stay in Japan?
もちろん、私は在留資格を持って、日本に在留しています。
Of course, I am residing in Japan with residence status.
私がこの国に在留することができる期間は限られているんです。
I have a limited amount of time to stay in this country.