I just write text docs in Emacs and use Git for version tracking. If he wants real-time tracking it's possible that one of the collaboration editors can do that and I might look into it. In the extreme you could videotape yourself working and typing, and maybe saying out loud what you are doing. I think the purpose is to show that you were doing the work yourself rather than prompting an AI.

I guess there are also software screen recorders and maybe there's a way to log X events (keystrokes and mouse movement) if that approach is ok with both of you.