Tereza May’s solution is just more political rhetoric.: “Landlords and doctors must verify if the applicants are in the UK legally” without any instructions on how to prove it.
The simple solution is to set up a sytem similar to one that has been in use in Canada for many, many years, and require all legal immigrants and citizens to obtain an ID card, preferably with photo ID and a chip, which must be presented to employees, landlords, hospitals, employment agencies, welfare offices, etc. prior to obtaining, services, benefits, employment, etc. The ID must be kept on the person at all times IF they need employment or any government services.
Crooks have found ways to circumvent the system by using the names of dead persons, etc., and obtaining documentation that way, however, generally speaking the Canadian system works well. It is commonly known that many businesses will employ persons who are illegals or on welfare “off the books” but inspections and hefty fines can alleviate some of these practices.
NOTE: In the USA any serious crackdown on illegals would shut down Las Vegas and wreck farm harvesting, where, apparently the majority of the service workers are illegal immigrants as well as deny many people who ought to know better, the use of cheap servants.