Coronation
An unpopular solution may be to insist on ID cards to be carried at all times. This would state how long there allowed to stay, if legal here etc.
But many people wouldn't be happy paying for ID cards, and do we have the police /border resources to carry out checks?
I do think it's easy to disappear into the country and this may be a factor in people wanting to come here.
Of course there are enough border force to check travellers, every time I travel into or out of the UK my passport is checked. There might be a few illegals evading checks hiding in lorries, and a few arriving on a beach undetected, when they are eventually found they all try to claim asylum anyway.
I’m sure more than half migrants have been Persecuted in their own country, it’s not hard to upset the government in at least half the countries in the world, wrong religion, wrong tribe and most can speak English as a second language, of course they want to come here, many have relatives here as well.