How will border control at the airport distinguish between EU citizens who are allowed to come in, and those who need to fill out some paperwork and will be kicked out if they overstay their implied visa / waiver?
You don't seriously expect this government to actually think through their proposals, do you? /s
More seriously: permanent residency as a result of exercising treaty rights comes with a document, so some will carry that. For everyone else, you would hope the government will put in place some sort of registration scheme that is easier than treaty-rights (which currently requires 5 years of continuous settlement in UK).
As for prosecuting people who overstay... eh. They are already pretty overworked and incapable of tracking most illegals, I can only imagine things will get even worse after they add hundreds of thousands new cases. Like in the US, they will compensate for this structural deficiency by being extra-brutal with the poor few they do catch (like they already do). This will drive more and more people underground, which will do wonders for crime syndicates, human traffickers and the like.
