Xiaomi
Allow or Block Display over Other Apps auf Xiaomi
On stock Android this permission is called Display over other apps. MIUI renamed it Display pop-up windows while running in the background, but it controls the exact same behavior.
Apps like Messenger or your default Messages app rely on it to draw chat bubbles or call screens on top of whatever you are currently using. If the permission is off, the app can still send a notification, but it cannot overlay content on your screen.
This setting is managed per app, so one app can have it enabled while another stays blocked. That matters if you only want floating bubbles from one messenger and not from every social app installed on your phone.
To change it, go to Tap on Settings ➔ Tap on Privacy protection ➔ Tap on Special permissions ➔ Tap on Display over other apps ➔ Choose an app ➔ Enable or disable Allow display over other apps.
Turning this permission off for apps you do not use for messaging or calls reduces unwanted overlays and can slightly help battery life, since the app no longer keeps a background window ready to draw.