(This is the way we've been playing for this campaign, I think it has worked OK so I'm writing it up.)

Instead of rolling for initiative and sticking to a turn order, play just flows as part of the normal conversation, just like when you're exploring the jungle or talking to an innkeep. It's not a special "combat mode", you could be talking and then shoot someone and then look inside a chest and then hit someone and then talk some more. It's seamless.

Normally the side that is under attack is the side can make their attack actions first, and deal damage first, unless there's an ambush, or violence is unexpected.

I'll keep track of actions so that everyone gets the same amount of actions.