I don’t think that is possible with Asana natively. Maybe check out The Asana automation bible: all the triggers and actions from Asana, Zapier, Flowsana, Make.com for alternatives.
I usually advise my clients to be really careful about having such a system. You are basically duplicating the info, by having an “assigned” field in addition to an actual assignee, just like you would with a “completed” section. That is the best way to have data inconsistencies. Also if you can’t find an automation, everyone need to work more by doing actions twice.
Good luck!