I tried just adding a UINavigationBar directly to my subview (rather than using a UINavigationController.
It works fine, but the nav bar appears differently from how the built-in ones in Calendar and so on behave. It is not translucent no mater what settings I use in IB, and there is a visible "seam" around the edge of the nav bar that is not present in the built-in examples.
Are popovers do some special extra magic when the content view controller is a UINavigationController? If so, I wish Apple would document this.