Most popular browsers stop web apps when there is another full screen app in the foreground.
To fix that on Chromium based browser (Chrome, Edge, Brave, etc) go to chrome://flags and disable #calculate-native-win-occlusion.
Update 1: This flag is being removed, for now it can be temporary brought back enabling flags starting with #temporary-unexpire-flags, but not for long.
Update 2: The flag is gone. Head over to r/Granblue_en subreddit for a possible fix: Disabling Occlusion in Chrome 99+.
Update 3: Looks like the flag #calculate-native-win-occlusion is back in version 101+, but I would suggest using both the flag and the command line parameters.
To fix that on Firefox go to about:config and set full-screen-api.warning.delay to -1 and widget.windows.window_occlusion_tracking.enabled to false.
Chrome also suspends non-active tabs. Disabling flag #intensive-wake-up-throttling or extension The Great Suspender should remedy that, however it did not work for me at a first glance. Not sure about Firefox.