MARY, a Zimbabwean woman, first came to Britain in 2003. She flew in on a South African passport and immediately claimed asylum. Her claim was rejected, as the Home Office believed she really was South African. But, like many asylum seekers, she did not leave. Several years—and appeals—later, she was almost deported, but avoided it by cutting herself with a blade she found in her cell. She has no right to be in Britain, let alone to work or claim benefits. But nor is she likely to be deported. “I don’t really know what happens now,” she says.

Her options will narrow further as the result of a new immigration bill, which passed its second reading in the House of Commons on October 22nd. This will oblige landlords and doctors to check the immigration status of their tenants and patients, make it harder for illegal immigrants to obtain bank accounts and driving licences and crack down harder on sham marriages. Appeals against immigration and asylum decisions will be granted only on fundamental human-rights grounds, not procedural ones.

The bill is the latest in a series of government efforts to reduce immigration. It follows curbs on student and work visas and changes designed to prevent the poor from importing spouses. Immigration is broadly unpopular in Britain, the illegal kind particularly so. The opposition Labour Party not only backed the new bill in the Commons but has pledged to amend it by introducing even tougher measures. Among other things, it would ban employers from running foreigner-only shifts.

Nobody knows quite how many people live in Britain illegally. A plausible estimate by the London School of Economics in 2009 put the number at 618,000, around 70% of whom live in London. That study relied on census data from 2001, and had a margin of error of 200,000.

Politicians fixate on human trafficking, which is particularly offensive, both to law and order and to the people trafficked. Charity workers tell harrowing stories of children working as “gardeners” in Vietnamese cannabis factories and teenage stowaways abandoned at service stations on the M6 motorway.

But trafficked and smuggled entrants are rare—they probably make up less than a tenth of the total. And there are fewer today than a decade ago, when border controls were looser. The vast majority of illegal immigrants arrived in Britain legally and then lost their right to stay. These divide roughly into two camps: people who overstay their visas and failed asylum seekers such as Mary.

Though a few will have driving licences and bank accounts acquired legally, illegal immigrants of all sorts tend to live outside formal society, which makes it difficult for the government to reach them through tough laws. Mary lives with friends in an east London suburb and is supported partly by a British ex-boyfriend. Other failed asylum seekers drift from town to town, staying at the homes of established refugees, says Jude Hawes, an adviser at the Citizens Advice Bureau, a charity.

Britain’s large, lightly regulated labour market is a big draw for some. Those who work tend to be paid low wages in cash, although some will borrow or rent national insurance numbers. Bangladeshis can find jobs in curry houses; Australian backpackers sometimes work in pubs and hotels; Nigerians often clean offices. Illegal migrants from eastern Europe in particular find work with outfits that employ lots of legal migrants, such as Poles, informally. Outside a hardware store in Cricklewood, a group of mostly Romanian men—not yet allowed to work—idles, smoking cigarettes and waiting for construction jobs.

Shadow boxing

For such people, renting formally from letting agents is already almost impossible—few have the references and income required—so the government’s new rules are unlikely to have much effect. Indeed, even legal immigrants tend not to use normal letting agents, preferring to use the informal sector. Flats are often let legally and then sublet to migrants through online exchanges (to which English speakers get no reply). Stricter controls will, however, inconvenience legitimate landlords. The Residential Landlords Association grumbles that members will need to learn to recognise up to 404 different European identity documents.

The same may prove true of doctors. Unlike many European countries, Britain has no formal identity-card system, so checking the residency rights of patients will be difficult, not to mention unpalatable. But few illegal immigrants use public services much, says Myriam Cherti of IPPR, a think-tank. They tend to avoid registering with doctors or visiting hospital when they fall ill, for fear of being asked awkward questions. The government claims that illegal immigrants cost the NHS £330m ($535m) a year, though it admits that the figure is highly speculative.

Reducing the right to appeal will probably do the most to make life more difficult for illegal migrants. Home Office decision-making is notoriously sloppy: in the year ending in June, fully a quarter of asylum decisions were overturned. Around a third of family visitor visa appeals were successful until the right to appeal was abolished. Yet immigrants may try to respond by claiming that their human rights have been violated, or turn to costly judicial reviews, says Joanna Hunt, an immigration lawyer. Indeed, one perverse effect of fewer appeals could be an increase in the number of people living in Britain illegally.

That number is probably already falling, says Franck Düvell, an academic at Oxford University. The economic downturn may have deterred illegal migrants just as it has deterred legal ones. Tighter rules on family and student visas mean that fewer people are arriving who may then overstay. The number of new asylum claims has fallen by 73% since 2000. The backlog is gradually being worn down. Some illegal migrants marry European citizens; others finally win asylum. Some create family ties strong enough to stay under human-rights laws. And some push off. Last year 30,000 left Britain voluntarily, having breached immigration laws; another 15,000 were deported.

Of those who remain, most stay in Britain because it is still better than returning to their home country. Little is likely to induce Mary to return to Zimbabwe. After ten years in Britain, she now has a child, cared for by its British father. No one is likely to try to deport her again. Instead, she will wait for as long as it takes to acquire legal status. Her travails, the government hopes, will deter others like her from coming to Britain in the first place.