Earlier quoted context omitted.
On top of that, the BIS isnt even needed for global real time payments. A company like Wise (formerly transferwise) or any similar entity can just hold accounts in each country and if the local settlement is real time they can also do real time global settlement by just updating their db and sending the money real time in the receiving location.
This is how they all work including western union. It's not a secret but it's not really talked about either. No one pays FX fees on each transfer. Single large rebalancing transfers allows them to actually move money at optimal times and rates with bulk discounts. Then they can currency hedge and get larger spreads on the actual fx rate.
My assumption was that it usually comes down to finding someone else with money in both currencies (i.e. a large bank or government) and exchanging one for the other. Of course that's unsatisfying: it's not just turtles all the way down.
Ultimately if I'm running out of, say, USD, and have a lot of CAD, I have to to buy a bunch of something for CAD and sell it in USD. If you wanted a "zero banking" currency transaction I guess the way to do it is to park on the Canadian side of the border, buy imports to the US, walk across to the US, and sell them.
Or maybe there's a magic money shredding / printing machine that I didn't know about. I guess an international treaty could actually authorize such a thing.