El Capitan has been working flawlessly on my early 2011 MBP, so I recently installed it on my wife's late 2011 MBA, with disastrous results. I suspect my mistake may have been changing her user login password to something different from her apple (and iCloud) password, plus enabling Keychain in iCloud, which I had not done on my MBP.
The following steps seem to have stopped the login pop-ups.
Log out of iCloud account.
Go(+option)>Library>Keychains
Move Keychains folder to trash. This will delete all of your saved passwords, btw.
Restart. Relogin.
Do not login to iCloud account. (Fortunately, we sync mail, contacts and calendars via Google and not iCloud, so syncing was not a problem for us.)
Go tell Apple to fix their ****.