Okay, let's say there's two characters on my opponent's side with 3 str each. Threat from the North lowers their str to 1. I put poisoned wine into play on one character, so he's discarded due to Threat from the North's secondary effect. Can I trigger Poisoned Wine's any phase ability, and move Wine to his other character?

No. Once Poisoned Wine gets rid of the character, that interaction needs to FULLY resolve (which will put Poisoned Wine in the discard pile) before any further "Any Phase" actions may be taken.

Put another way, the only thing that can interrupt another action is a Reponse: action that uses the word "save" or "cancel".