Explain a complex event handling case in AngularJS where controllers interact with services to update application data.

In a complex event handling case in AngularJS, controllers can interact with services to update application data by combining $emit and $on for bidirectional communication. Controllers can emit events, and services can listen for these events to perform data updates. Conversely, services can emit events, and controllers can listen for updates. This bidirectional communication ensures that controllers and services can seamlessly cooperate to keep the application data in sync. Understanding this advanced event handling scenario is essential for managing complex AngularJS applications.