AS BRITAIN considers its future outside the European Union, its main target for a post-Brexit trade deal is the United States. The prospect of opening up the American market is an enticing one. Yet some in Britain worry about what might be demanded in return. Perhaps most emotive is the suggestion that America could negotiate greater access for its companies to the National Health Service. Jonathan Ashworth, the shadow health secretary, has warned that a trade deal could lead to “stealth privatisation” of the NHS. “It is beyond belief that our prime minister is bartering away our health service in her desperation for post-Brexit trade deals,” he recently said.

Private involvement in public health care has been growing in Britain for some time. After 1997 Tony Blair encouraged private providers to take on more NHS work, to help the service cope with increasing demand. That trend has continued. Government funding to the private sector reached £8.7bn ($11bn) in 2016, or 7.6% of the NHS’s total revenue budget.

Almost all American investment in British health care so far has been in the private sector, through acquisitions. Acadia, a Tennessee-based health-care giant, now owns the Priory Group, a chain of posh drying-out clinics and mental-health centres. Hospital Corporation of America owns several private hospitals in Britain, including the Portland, a favourite place for celebrities to give birth.

The involvement of foreign companies in providing services to the NHS, meanwhile, is marginal. Some believe that after Brexit the government might be tempted to open things up if it thought such a move could buy Britain better access to foreign markets. It would have leeway to do so, since it alone would be responsible for conducting the trade talks. “There is no involvement of civil society, trade unions or parliamentary oversight at all,” complains Mark Dearn of War on Want, a campaigning charity. Others point to a precedent in the form of concessions that Australia made regarding its scheme for public drugs-purchasing, as part of a free-trade agreement with America in 2004.

It is conceivable that American firms might be allowed to tender for, say, a regional ambulance contract or community health services, just as European companies can, says Nick Fahy of Oxford University. British firms such as Virgin Care already do. But the government is unlikely to allow foreign firms to run large parts of the NHS such as hospital trusts, he believes. In the only example of its kind so far, Circle Health, a British company, took over management of Hinchingbrooke hospital near Cambridge in 2012. The experiment was not considered much of a success and ended after three years. It is not likely to be repeated soon.

And there is a prior question: would American companies actually want to invest in the NHS? Public health-care systems in western Europe are among the only ones in the world where the money involved is enough to make investment worthwhile for American firms, if they could gain access. An American company that could demonstrate the ability to run a British hospital might persuade countries like Germany and France to open up to more outside investment. That is something European governments would not budge on during talks over the Transatlantic Trade and Investment Partnership, a now-doomed trade deal that America and the EU had explored in the pre-Trump era.

At the moment, however, there is little enthusiasm “in America or anywhere” to invest in British health care, says Richard Murray of the King’s Fund, a think-tank. Government spending on health as a share of GDP is lower in Britain than in most of western Europe and the gap is forecast to increase. Many NHS trusts are in deficit. And dealing with the NHS can be a messy and frustrating business, far removed from the cash-rich American health-care machine. “Britain is just not a very attractive market,” says Mr Murray.

His main concerns about a trade deal are that it could give new powers to American firms to make legal challenges against the NHS’s monopoly, and that it could give new rights to foreign investors to seek compensation for lost profits if government policy changed. Neither seems likely, he says, because the government knows how unpopular such moves would be.

America will negotiate hard. But given the complexity, unprofitability and political sensitivity of dealing with the NHS, it seems unlikely that the feared “stealth privatisation” will take place. American companies will continue to invest in private health care in Britain—and leave the NHS to stagger on.