Why didn't the dragonballs work after Kami fuses with Piccolo?

As far as I know it, Kami and Piccolo were connected one a spiritual level since they were originally one person.

When Piccolo dies, Kami dies. When Kami dies, the Dragon Balls no longer work.

But when they fuse into one being again, the Dragon Balls still deactivate.

How does that work?