> The Nextcloud iOS app does it.

Searching for "nextcloud ios background sync" shows a whole bunch of forum posts and bug reports about it not working well unless you have the application open.

One issue (https://github.com/nextcloud/ios/issues/2225) been open since 2022, seems to still be not working properly. Another (https://github.com/nextcloud/ios/issues/2497) been open since 2023.

For something that works well it seems like a ton of people have a lot of issues with it. Are you sure you're on the latest iOS version? Seems like people experience the issues when they're on a later version.