Britain is forcing many of the world’s most notorious tax havens to reveal their inner secrets, after prime minister Theresa May’s government caved to an opposition amendment that will make the UK Overseas Territories open up their corporate books.

The amendment requires tax havens like the Cayman Islands, British Virgin Islands, and Bermuda to publish registries of who actually owns companies registered there. These islands now provide almost no meaningful corporate ownership information, resulting in abuse by kleptocrats, gangsters, and tax evaders—as found in the Panama Papers leak, where around half the companies exposed were formed in the British Virgin Islands.

Anti-corruption campaigners lauded the move. “The UK’s tax havens have featured in countless corruption and money laundering cases—ending their corporate secrecy will throw a huge spanner in the works of corrupt dictators, tax evaders and organized criminals,” Naomi Hirst of Global Witness NGO said in a statement.

There’s no exact estimate for the wealth held in the territories. Boston Consulting Group puts the amount (pdf, p.16) in the Caribbean and Panama as a whole at $1.3 trillion, more than 10% of the total estimated wealth held offshore. The news was met with outrage from the tax havens, which—alongside the financial sector—have bitterly opposed the UK forcing transparency on them, making arguments that range from this being a colonial imposition of power, to forcing the islands to lose their only significant industry.

Supporters of the amendment framed it as the first step in making transparent ownership registries the international norm, with the ultimate aim of ramping up pressure on the US to follow suit.

This is a long way from the end of offshore secrecy, however. Investors who want to keep their identities secret will simply up sticks to another tax haven, says Alex Cobham, chief executive of the Tax Justice Network, a pro-transparency advocacy group.

Secretive US states like Wyoming and Nevada will pick up business, he predicted, perhaps along with UK crown dependencies Jersey and Guernsey, which won’t be hit by the legislation. If businesses are happy to move to avoid transparency, it will be tricky to push the US in line with Europe’s wish for corporate ownership registries to become the global norm, says Maya Forstater, a visiting fellow at the Centre for Global Development: high levels of secrecy will become “a competitive advantage for US jurisdictions.”

Cobham says it’s possible that some of the targeted UK territories could declare independence—though doing so would mean losing one of the most attractive things they offer investors: access to Britain’s trusty judicial system. So, it’s crucial the UK government helps them transition, he says: “To put this through as an imposition without any related plan for significant financial support risks imposing real costs on the poorest people in those territories who are already suffering from the inequalities that come from being a financial secrecy jurisdiction.”