Any Remote Desktop software to remote into an iOS device from a computer?
If you consider a non-jailbroken iOS device, there's basically two types of remote software possible:

The first type lets you remote in to see what's on the screen in "real time". It works across apps, and even when you're on the home screen, in control center, etc. The main drawback is that you can only watch what's happening - you cannot control anything. A common system that implements this scheme is TeamViewer.

The second type lets you remote in to see what's on the screen in "real time" and also allows remote control (and possible various other actions such as "painting" on the screen, popping up messages, etc). The main drawback being that it only works inside a single app - as soon as you exit the app, the remote connection stops. Several vendors offer SDK based solutions for this type of remoting - but you have to be an app developer to embed it inside your own app. You cannot apply it to any random app you have installed.