We have a hybrid setup with Exchange 2013 and O365, dirsync and ADFS are configured and working correctly, end-users have Outlook 2013 running on 2008R2 RD Servers. Mail-flow and migrations are working correctly. O365 homed users are not prompted for credentials when opening Outlook 2013.
The problem arises when on-premise homed users try to access an O365 user's calendar. Outlook prompts for credentials when connecting to the calendar, when the user enters their credentials the calendar opens as expected.
The O365 calendar permissions have the default set to reviewer, the same as the on-premise users.
The OutlookAnywhere authentication settings have been set to Negotiate following a suggestion from here. As far as I can tell it has had no effect on the problem.
The OrganizationRelationship is configured at both ends of the Hybrid connection, following a suggestion here. The FreeBusyAccessLevel is LimitedDetails.