They file for an ITIN. An employer can help them file, also. It's a number for tax ID purposes that isn't an SSN since they can't get an SSN. The IRS has had the ITIN program avaulable for undocumented immigrants since the mid 90s.

There’s a few possible ways I can think of.

For one, the bank account doesn’t have to be in the “fake” persons name. If you use a stolen identity to work it won’t be any issue except unless the real person sees something weird on their taxes.

Also consider that in many cases there are very common names for some immigrants. “John Smith”

You can also get a job and work for a few days. Even if you are illegal you still have a right to be paid and they may not verify in time, or at all. Many states don’t require employers to verify the SSN. https://en.wikipedia.org/wiki/E-Verify

Also you could work as part of a crew. The head of the crew negotiates with the employer on the basis of the documented employees but then splits the earnings over the crew themselves.

You can apply for a taxpayer identification number. This lets you (and presumably your employer) pay taxes even if you are undocumented. https://www.americanimmigrationcouncil.org/research/facts-about-individual-taxpayer-identification-number-itin

Naturally you could also work for a cash business on the black market.

They can. The US allows tax payments from non US people in the country illegal or not. We did it.

FYI. One of the questions that gets asked when applying for citizenship is if while in the country illegally did you earn any money, how much, and proof of paying taxes on that money. If you didn't earn any money while in the country then how did you survive, etc. Most would answer via working legal spouse to the latter.

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