For some definition of “server”, yes.
https://www.raywenderlich.com/5817-background-modes-tutorial...:
”Your app is only allowed to keep running in the background in very specific cases. For example, these include playing audio, getting location updates or fetching the latest content from a server.
If your task does not fall into these categories, backgrounding may not be for you. You may even find yourself with an App Store rejection if you try to cheat the system by using background modes outside the realm of their purposes, so consider yourself warned!”
If you read that, your ‘server’ either will only run for a finite time determined by the OS (in practice 3 minutes), or only run at highly granular times determined by the OS (”the system uses usage patterns to determine when to best fire off a background fetch. For example, if your user opens the app at 9 AM each morning, it’s likely that a background fetch will occur sometime before that time.”)
I think a X11 server that tracks updates for those three minutes of inactive time, and has a window manager that requests window updates where needed when the app becomes active could work, though.
(Disclaimer: these restrictions get relaxed over time on iOS. I may have missed some recent improvements)