Back in the 1980s, some Italian cafes employed a similar system. As a customer, you'd place your order at one window, and the clerk would give you a receipt/voucher. You'd then step to a second window, and hand over the slip of paper before stepping to a third window where you'd collect your coffee. I recall, though these are hazy old memories, at least one morning where it was the same person at each of the three windows.
Also, I once did the same PRC consulate dance in SF!