Log to Normal Screen in AltScreen Mode

**Is your feature request related to a problem? Please describe.**

I have a TUI app that also runs a server in the background. The server should be able to log something, but this currently breaks the AltScreen TUI

**Describe the solution you'd like**

An option to log to the normal screen. Such that when quitting the app, the logs from the server remain

Start by tracing how AltScreen mode manages the terminal screen and how applications currently emit logs. Define the option's behavior for logging to the normal screen, including what remains visible after the TUI exits, then identify the tests needed to verify both screen modes.

Written by the indexing model from the issue text.