Aren't employers in most jurisdictions required to verify work eligibility and report income? Does "pseudonymously" just mean that they these employers commit to supporting your alternative identity when engaging with the public, the way publishers might handle a pen name? Or is this something different? I'm probably not the only person who will wonder about this, so it might help to put the answer as text on the sit…
Basically, they kept as low as possible the number of people who knew their real identity. That person or persons would verify employment and file the appropriate paperwork.