After some research, I came across these two forum posts, describing exactly the same behaviour: https://developer.apple.com/forums/thread/778184 and https://developer.apple.com/forums/thread/772999. The answer in both was to enable all interface orientations for iPad.

I tried that via the project settings in Xcode:

Alas, selecting all orientations for iPad did not work.

But then I remembered in our app we also (for reasons) define this set of options programmatically, via the AppDelegate. I applied the same changes there:

func application(_ application: UIApplication, supportedInterfaceOrientationsFor window: UIWindow?) -> UIInterfaceOrientationMask {
    switch UIDevice.current.deviceIdiom {   // `deviceIdiom` is our own property for handling device idioms
    case .phone:
        return .allButUpsideDown
        
    case .pad:
        return .landscape
       
    case .mac:
        return .all   // <-- Return `.all` here!
    }
}

And, voila, we have content in popovers on Mac Catalyst!