The store will continue to trade as Allders and be run by Tillman's business partner Andrew MacKenzie. He has been appointed chief executive of Allders (Croydon) Ltd, which will operate the store.
Minerva owns the freehold and Tillman (pictured) is believed to want to work with the property company on redeveloping the site.
Allders Croydon is the UK's third-largest shop and has been trading for almost 150 years.
The deal ends speculation that Minerva was about to sell the lease to John Lewis.