Puppet overlapping with existing city, selecting which tiles puppet can work?

I've recently puppetted a nearby city, who's 3 square radius overlaps with one of my existing cities. If I go to the existing city "citizen management" screen and click on the tiles, it seems the existing city gains "ownership" of these tiles, i.e. the city automatically applies citizens to work them if appropriate. However, when I go to the puppet screen, there's seems to be no way for the puppet to take ownership of the tiles back, as a result the puppet will not work these tiles once the existing city no longer does.

Is there anyway to return these tiles to a puppet, or is the transfer of tiles between an existing occupied city and a puppet a one way process?