Troubleshooting Shelf Tabs in Autodesk Maya

Autodesk Maya has many useful tools, some of which don't always work the way you want them to.  One super-useful tool is the Shelf Editor, which allows animators to select parts of an animation rig quickly and efficiently.

But a common problem occurs when you import an animation rig into a new shot, and all your carefully created Shelf Tabs don't work anymore.

So, what is the solution to this annoying problem?