People traded internationally before the invention of currency
The pop idea of barter prior to the introduction of currency is an invention, and it has little to do with how people actually traded. The idea that people were limited to on-the-spot exchanges of say two chickens for a sword and a cabbage is completely ahistorical. In fact, people were making complex, and temporarily disjoint trades long before this idea was formalised into currency. Debts were simply recorded and settled in terms of goods rather than abstracted into currency. This trading was smaller scale than it could be with currency but not directly different in kind.
Currency was a formalisation of existing practice
Currency didn't spring into being out of nothing, it came into being because it formalised trades that were already being made. The first currency was precious metal, traded by weight, which was later stamped into coins of known weight. Ancient Japan used rice as an accounting unit since it was the staple on which other things depended. 1006
Conceptually, goods can be thought of as a form of currency with a complex exchange rate between them, and so the conceptual step from trading in goods to trading in currency is not a big one.
So, what does this mean for a world without currency?
How could we scale up these pre-Currency practices to a global economy on the modern scale? The answer lies in the concept of debt. A debt is nothing but a promise to deliver, and there's no reason that these can't be traded. Thus, I, a chair-maker take to you, a grocer, a promise to make a chair, I trade this promise for my weekly shop. But the promise is worth more than the weekly shop so you also give me a promise for 8 gallons of beer that the local brewery paid with. I can then "spend" that promise elsewhere.
At first, this is just barter, but we can scale it up to a global level through the introduction of markets and clearing houses. Just as today stocks are traded on stock markets, I can take the beer promise to the clearing house, and find it's value in anything else by finding what the market will offer and trade it to a promise for whatever thing I want that I can then take to a local provider to get that thing. This requires that providers permit interchange, which will require legal controls, guilds, or other co-operation, but - remember - this is exactly what we're doing right now except that we abstract it into currency so it's operation is not implausible.
Overheads will be higher, there are more middlemen, and it introduces new complexities but the system should be capable of scaling up to modern industrial scale. The introduction of computing power and mass communication actually makes it much more practical.
The difficult question remaining is why on Earth your civilisation didn't make the small leap from here to currency. 130