Great question!

A dry exchange is actually a matter of historical debate. As Raymond de Roover puts it, “the nature of “dry exchange…has given rise to a good deal of confusion; and to most people the meaning of the terms remains obscure.” Different scholars have different opinions as to what “dry exchange” was, based on conflicting sources.  

As best I understand his article, dry exchange involves a legal or conceptual fiction, whereby interest rates (forbidden as usury) were concealed by adding them on to exchange rates. To quote the article:

“In view of the existing organization of the money market, only merchants who had connections abroad could take up money by exchange, because the taker had to be represented in other places by an agent, a partner, or a correspondent who would accept and honor his bills. If this representative did not have sufficient funds to meet maturities, he could always raise money by drawing on his principal. In any case a merchant who had no such connections was seriously handicapped. He was shut off from the money market, unless he could make special arrangements with a deliverer. The purpose of dry exchange was to make such arrangements possible. “Dry exchange” was merely a device by which a local loan was disguised under the form of an exchange transaction.” 

In other words, if I’m an Italian banker who wants to buy wool cloth in England, I need pound sterling to make the transaction. But if my bank isn’t particularly well known on Lombardy Street because we don’t normally do a lot of business in England, I could be out of luck because no one will take my bill  of exchange. 

However, a “dry exchange” would allow me to get a loan in pound sterling from an English banker. The English banker would, obviously, want to make a profit from this loan, but isn’t legally allowed to charge interest. So instead what happens is that I go to the banker and fill out a bill of exchange from Florins to Pounds Sterling where the banker is both payer and payee (this is what makes it fictional, he’s basically paying himself), and then the banker fills out a second bill of exchange from Florins to Pounds except for that it’s payable to me, which I can then take to a cloth merchant. He has essentially loaned English pounds to me which I then turn around and use in another exchange. 

The main difference between the two bills, besides the payers and payees, is that they would have different exchange rates between Florins and Pound Sterling, which would net the banker a profit on the exchange. However, legally speaking, the banker could say that he hadn’t actually loaned money at interest, but rather conducted two currency exchanges at variable rates.