datasync-global-dg makes the F5 to go offline

The datasync-global-dg device group is a special device group. When you first provision ASM it is created. You may have noticed at this time that your CPU went to 100% and the device remained unresponsive for up to five minutes. When you first add another BigIP to your trust you should sync the datasync-global-dg from one unit to the other. It doesn't sync any of the normal things you would expect, but rather shares cryptographic information used by the ASM.

From https://support.f5.com/kb/en-us/solutions/public/16000/500/sol16509

Impact of procedure: During the synchronization, the devices receiving the sync may go offline for approximately 3 minutes. After the synchronization is complete, the receiving devices should return to an online state. F5 recommends that the synchronization be initiated from the device that is currently active to avoid potential traffic disruption.

As such the behavior you are describing is expected (especially if you have turned on autosync for this device group). You will want to read SOL 16509 for a better understanding. This also explains why you cannot select this device group under: Options -> Application Security -> Synchronization -> Application Security Synchronization. It is not a sync group in the traditional sense.

If you do not have autosync and are not syncing this device group manually you may wish to open a case with support and have them take a look.