My items on the desktop are not synchronized in real time while the mobile device is connected.

Presently BirdieSync doesn't offer a real time synchronization on PC side (although it does on mobile side). If modifications are done in Lightning after you have connected and synchronized your mobile device, you must then launch a new synchronization manually, before disconnecting your mobile.
This limitation should be addressed in a next version.

You can choose a particular synchronization period in BirdieSync synchronization options (File/Options/Synchronization frequency).

You don't need to disconnect your mobile to manually relaunch the synchronization. You can click on the "Synchronize" button in BirdieSync window, or you can select the "Start synchronization" command in the context menu of the BirdieSync icon (the green one) in the task bar.