Just require them to show valid ID. Having people produce the proper documents and then verifying them isn't the same as keeping track of people.
without giving the government the ability to track people
I think the government already tracks you, at least a little bit. Your income is reported to the IRS, as is the interest on your bank account and mortgage. Also, universities have to send a notice to the IRS regarding every student, because of some very limited tax deduction scheme that Algore invented; even if you're not eligible (and most people aren't), the notice is sent. Waste of paper.
All for it, but what constitutes "valid ID"? A lot of illegals can buy stuff that passes on a quick glance by an employer looking the other way. Do national standards for tamperability constitute a "national ID card"? I don't think so, but some people here would holler loud and long if we put counter-proofed computer chips in state ID cards.
I just got a duplicate Social Security card in the mail about three and a half months ago, and it seems frightening easy to duplicate or alter. I'd love to see it meet the standards of even the most lax state drivers license! Your point about the government being able to track you through the IRS makes me wonder if we should change the timing of the tracking. Instead of letting the government know where you worked by January 31st of the next year, what if they had a way to know where you were working when you got a job? That might discourage a lot of illegals.