You become quasi landlord!
Well, the tenant who is staying behind simply steps into the shoes of the main landlord and becomes the lame tenant's principle landlord and as such can sue to recover for all unpaid lease in an Unlawful Detainer action or forceful detainer action, depending on the state you are in. But they are basically the same action but referred to by different names.
But there is a CAVEAT! You must make a concerted effort to mitigate your damages by searching for a new tenant and cannot simply "rest on your claims" for the entire default period.
fredrikklaw