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If you meant a prerecorded voiceover seemingly reacting to things in real time, that's The Tape Knew You Would Say That. It covers actions too, so it's also "The Tape Knew You Would Do That".
I'm otherwise not clear on what "an obviously post-recorded voiceover responds to action on-screen as though it were being recorded on site" means. Are they just a Narrator? They might be a Lemony Narrator if they're being particularly weird about it.
Edited by KD

There's a thing I've seen often on Binging with Babish and The Backlogs among others, where an obviously post-recorded voiceover responds to action on-screen as though it were being recorded on site - saying "Whoop" when dropping an ingredient, or offering a "Glurk" when an in-game avatar is smashed by an enemy's hammer. Is this a Thing?
Edited by hobbitguy1420