Practically-speaking, we cannot erect an impermeable wall and still, at the same time, do effective and prosperous business with the outside world. Nor do we want to open the flood gates and for all--past and future--trespassers to have sudden immunity once inside our border.
One solution is to have a good internal tracking system, one that seeks and (prosecutes) deports illegals after exacting at least a nominal penalty. To argue that this approach would "require" a national ID is a shallow reaction because such a system would not need a national ID---identifying, finding and arresting illegals can be done with current state-of-the art technology and one more thing:
An effective, ethical, competent and loyal FBI.