iCloud keeps asking for password on iPad, iPhone & Macbook Air
I have an iPad Mini, iPhone 6 and Macbook Air, all of which get continual prompts to enter my iCloud password. All have fully up to date OS. On the iPad, I've done a full erase and restore, to no avail. Since then, I've also signed out of iCloud and signed back in on the iPad, I haven't tried that yet on the other devices. My iCloud user name is a @me.com format, so I've made sure that all log in places (FaceTime, iTunes Store, Messaging, etc, use that log in, even though I have an @icloud.com email too. I disabled 2 factor authentication, and App Specific Passwords for Messages, and FaceTime.
It seems to have something to do with email in particular. On the Macbook Air, using Apple's Mail program, I get a message on the main window that login has failed, and then the iCloud account is taken offline. I have to re-enter the iCloud password and/or quit and restart mail to get it to work.
Another thing pointing to the problem being with email is that I turned off all Push email for the iPad, and the password prompts stopped. Then, when I manually try to get mail, the prompt pops up again.
I'm about out of ideas. One thing I haven't done is to reset to a new password, or to sign out on all devices, then add them back, one by one.
Any help would really be appreciated.
Thanks,
Dan