How to integrate core data and cloud storage?

Apple released a new way to access documents on iOS and iCloud via UIDocumentBrowserViewController. Though the Xcode template only allows UIDocument types.

Core Data is great for managing document data, but the NS/UIManagedDocument types are apparently the ugly stepchild when it comes to iCloud, syncing and support in general throughout iOS.

Is there any way (or what would be the best approach) to use both core data and iCloud document syncing? I want to use core data to store the user data but also allow the user to sync its files across all their devices.

A custom NSDocument with a custom core data stack? A filewrapper that manages a sqlite file?

Can't there be only one sqlite file and isn't iOS copying Document files all over the place?