It’s working now. Interesting that AppleCare didn’t know what to do. The senior advisor was sending me on a wild goose chase. Anyway, what fixed it was logging out of iCloud, and then logging in again, on both computers. I read that somewhere, a suggested maneuver for trouble shooting iCould problems in general.

So I’m using Back To My Mac screen sharing between my iMacs and LogMeIn for access to my iMacs from my iOS devices. I agree with you, it’s the way to go. By the way, LogMeIn also is good for accessing Windows (which I run in VMs) from my iOS devices.

Thanks.