LONDON, June 24 (Reuters) - The British government is set to impose tougher than expected trade caps on steel, limiting imports from Vietnam, South Korea and Algeria to better defend its domestic supply from the fallout of a global trade war.

Britain has been trying to shield its domestic industry from a global oversupply of steel, and has taken several measures to support it, including negotiating the removal of U.S. tariffs - yet to fully take effect - and taking control of British Steel to prevent it from shutting down.

But the global upheaval to trade from Trump's tariffs and the precarious state of the steel sector has led to increased scrutiny of Britain's own trade defence regime.

Industry body UK Steel had said that measures proposed by the Trade Remedies Authority (TRA) in May that capped how much of certain kinds of steel could be imported would do little to defend the industry, as the levels of the caps were too high.

On Tuesday trade minister Jonathan Reynolds wrote to the TRA to say he intended to "reject the recommendation and to take a different decision" and go for lower caps instead, due to come into force from July 1.

"The reason is to ensure the overall effectiveness of the UK's steel safeguard measure for domestic producers whilst balancing the need for security of supply for the UK market," the letter seen by Reuters said.

The TRA initially recommended that three categories of steel - UK imports of which are dominated by Vietnam, South Korea and Algeria respectively - would be subject to caps that mean each country can only supply 40% of the residual quota. Any imports beyond that will be subject to a 25% tariff.

Reynolds said that the cap would instead be 20% in the category of steel dominated by Vietnam and 15% in the categories dominated by Korea and Algeria, restricting how much steel those countries can import into Britain.

UK Steel Director Gareth Stace said it was a "tremendous outcome," and a demonstrated Reynolds' commitment to steel.

"The measures will reduce the pressure of steel diversion from the US and EU and prevent countries that flood international markets with unsustainably cheap steel from swamping the UK and driving our steel manufacturers out of business," he said.

The decision is subject to final confirmation after World Trade Organization (WTO) members share their perspectives, Reynolds wrote.