I found a solution, though "work around" is probably a better description:
- Remove the storyboard (or XIB) from any Target Membership;
- Do a clean build (and clean the build folder as well);
- Close XCode;
- Delete the app from the device/simulator;
- Delete the contents of XCode's DerivedData directory (see Preferences);
- Rebuild and launch the app, navigating to the area where the storyboard/XIB is expected (this should cause a crash);
- Re-add the storyboard/XIB under Target Membership;
- Rebuild and launch.
Step #6 is the most important. If the app doesn't crash it means that you have a cached copy of the file somewhere, which is no good. I have no idea exactly why this fixes things, though I wish I did.
After this, things work swimmingly (using Base Internationalization or not).