Final answer:
The establishment of their own political and military structures signaled the independence of the barbarian kingdoms from Roman imperial authority. Examples include the Visigoths sacking Rome and the conversion to Christianity by barbarian kings.
Step-by-step explanation:
The fact that most clearly signaled the independence of the barbarian kingdoms from Roman imperial authority was the establishment of their own political and military structures. The barbarian kingdoms formed their own governments, laws, and military forces, separate from the Roman Empire. This demonstrated their autonomy and independence from Roman control.
For example, when the Visigoths, led by their king Alaric, sacked Rome in 410 CE, it highlighted their ability to act independently and assert their own authority. They were no longer subordinate to Rome and could exercise their power against the once mighty empire.
Another example is the conversion to Christianity by many barbarian kings. This allowed them to participate in the network of civilized societies and engage in trade and diplomatic relations with the Byzantine Empire and other Christian kingdoms. It further solidified their independence from Roman imperial authority.