When I was younger, I once imagined a universal court which would be above all supreme courts, but without any government backing or any force to enforce.
The sole criteria it would have is that the case before it has been studied by supreme court of any nation, and ruled in violation the principle this court observes. This court would then study the case and then rule punishment to the accused AND penalty to the erring judge of the supreme court. The enforcement of its will shall happen in the good old American way of bounties and bounty hunting.
It was quite detailed, but the relevant crux here is that enforcing justice doesn't need dedicated manpower or government backing.
However, a specified court/judiciary is definitely required for a legal recognition.
No. I am describing a global system that goes beyond state.