Investors in the new bonds were primarily tax-dodging Westerners, but they also included kleptocrats and organized-crime groups. By making the bonds anonymous, portable and redeemable anywhere, the London bankers deliberately made it easy for buyers to hide their identity.

This is the original sin that explains why the offshore system has remained so durable. The privacy it allows, either through shell companies or anonymous bank accounts, is as useful to regular businesspeople as to fraudsters or kleptocrats. Any effort to crack down on, say, mobsters who stash their money in New York real estate also inconveniences corporations seeking to minimize their taxes, and thus triggers concerted opposition.

Offshore often happens right at home. The laws in Delaware and Nevada offer companies similar benefits as the laws in the British Virgin Islands. In fact, according to one in-depth study of companies that create companies, it is far easier to set up a corporate structure without being asked to confirm one’s identity in the United States than it is in notorious tax havens, like the Cayman Islands, Jersey or the Isle of Man. The Russian arms dealer Viktor Bout is just one of many criminals to have hidden behind Delaware’s shell companies. Mr. Trump has used corporate structures in that state to hold assets, including a Scottish helicopter; Hillary Clinton, to receive speaking fees.

But what’s good for large corporations and rich individuals can be disastrous for society. Experts at the F.B.I., Britain’s Serious Fraud Office and other leading investigators of grand corruption say shell companies are the primary obstacle to bringing the world’s fraudsters to justice. One Miami lawyer told me they are like armor plating: Shell companies own other companies across multiple jurisdictions, creating layers of corporate plating so thick it becomes all but impenetrable.

Law enforcement in the United States leads the world in investigating and prosecuting corruption, but the country still lags badly in terms of transparency, despite FinCEN’s new powers. Britain has a freely accessible, central online register of all companies, and anyone who creates a new company has to declare its actual owners. The system is flawed, not least because the information provided is not checked. But it’s a good first step toward ending the abuses enabled by anonymous companies. America should adopt something similar.

The British Parliament is also considering implementing “unexplained wealth orders,” which would force criminals or foreign officials to justify how they came to own luxury goods or expensive real estate. Those who cannot prove they secured the financing legally would see their property confiscated.

These two measures alone can’t be enough to reveal the true owners of the anonymous money that is buying up so much of the West. But they can go a long way toward depriving dictators and thieves of a haven for their dirty cash, while helping officials in democracies regain control over who is buying the countries they were elected to run.