The App Store won't recognise verified Apple ID

A friend has a problem on her iPad as she appears to have two Apple IDs.

On the iPad and under Settings, the iTunes & App Store shows her Apple ID as an iCloud email address. If she taps that ID to view the ID information it asks her for a password, which is accepted, and it goes into the Account window which again displays her Apple ID as the iCloud address. Tapping that again gives her an Apple ID Summary, and the opportunity to click an appleid.apple.com link to change security questions and other Apple ID details. Clicking on that link takes her to a login page on the Apple website where she can sign in successfully with her iCloud ID and password.

She then sees confirmation of her name, her Apple ID and Primary Email Address (the iCloud ID/address) which is ticked as Verified. She also sees an Alternate Email Address (a gmail.com rescue address which she set up last month) which is also marked as Verified.

So far so good.

The problem arises when she goes into the App Store on the iPad and tries to do anything (update or purchase an app). The Store gives the gmail.com address as her ID and asks for a password for that (which she does not have). Why doesn't the store have her iCloud address as her ID - as would appear to be the case from the other information given above?

Is there a simple way to have the Store use her iCloud ID?