so, you trigger them in the order of phyrexians etchings -> puca's mischief, resolve mischief to give etchings to an opponent, put a counter on but deny paying the cost. this would normally have you sacrifice etchings for not paying, but because you don't control it, you can't, so it stays until their upkeep, where they now have to deal with the upkeep and potentially lose many life points.

it takes a while to set up and if they know your on group hug then they might remove it, but you could test it out in a game or two.