Final answer:
The common challenge shared by the Transcaucasian and Central Asian republics is their political and economic instability during their transition from being part of the Soviet Union to independent states.
Step-by-step explanation:
The Transcaucasian and Central Asian republics share a common challenge in terms of political and economic stability. Both regions have experienced difficulties in transitioning from being part of the Soviet Union to independent states. This transition has resulted in challenges such as political corruption, economic instability, and conflict. For example, the Central Asian republics have faced challenges in managing their natural resources and competing in the global economy.