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.

On XP you can choose a particular synchronization period in BirdieSync options (Tools/BirdieSync/Options/Synchronization/Synchronization frequency)
On Vista, the mobile device is automatically synchronized every hour.

You don't need to disconnect your mobile to manually relaunch the synchronization. On XP, you can click on the "Synchronize" icon of the ActiveSync window while on Vista, you can for instance select the "Sync all" command in the context menu of the Sync Center icon (the green one) in the task bar.