Teeth
Bottle caps are a currency in Fallout. No more are being made, and you have to work to get more (go out in the wasteland and find them), so they fulfill the same criteria as something like gold (rare and hard to obtain).
What if the Alliance uses Skibidi teeth as an unofficial currency? They're in limited supply, and hard to obtain, thus representative of hard work. They're probably contraband (especially now the Alliance joined the Union) but units use them anyway.
Astro teeth might be more like Rai stones.
The ownership of a large stone, which would be too difficult to move, was established by its history as recorded in oral tradition rather than by its location. Appending a transfer to the oral history of the stone thus effected a change of ownership.
Also:
In one instance, a large rai being transported by canoe and outrigger was accidentally dropped and sank to the sea floor. Although it was never seen again, everyone agreed that the rai must still be there, so it continued to be transacted as any other stone.
When transacting the stones, they'd be like 'the stone at the bottom of the bay is yours now.'
The perceived value of a specific stone was based on its size, craftsmanship, and history. The value could depend, for instance, on whether a famous sailor brought it or whether people died during its transport.
That could absolutely be extrapolated to winning Astro teeth in battle!
Bonus possibility: Astro scrimshaw.