you can play the external identity in two ways, at the individual or TEAM level
It can be played taking the existing model designed in the Team identity or make a bold movement about perceptions.
It depends on what you want to achieve. Making it at the individual level, you ask the participants to write their names on a piece of paper. You put all names in a bag, mix it and invite one by one to pick and choose one name randomly. You do not disclose the name but built a model of the name. It is about perceptions, what the others know about me that I do not know, entirely in line with Johari window.
When you play teams, the identity can be based on the performance of the Team.