A family of Microsoft word processing software products for creating web, email, and print documents.
Hi James,
Based on your description, the Macs keep asking you to sign in each time when you use either machine. To narrow down your situation, please try removing the activation files and reactivate Office 2016 for Mac:
Go to <YourUserName> Library > Group Containers and move the following folders to trash
UBF8T346G9.ms
UBF8T346G9.Office
UBF8T346G9.OfficeOsfWebHost
Regards,
Tisky