Can I adjust status via EB3 if I entered the United States without authorization? No. If you entered the U.S. without being inspected and admitted at an airport, or at a land border crossing station, you cannot adjust status (get a green card) from within the United States via the EB3 visa. Some family-based visa categories allow you to apply for a waiver of your unlawful entry, but that waiver is not available for employment-based EB3 visas.
What if I entered without inspection, but then I applied for asylum? Asylum cases are very complicated, and success depends on the unique facts of each case. At some point in your asylum case, you might receive work authorization or an eventual asylum-based green card. But that does not excuse your unauthorized entry if you want to apply for a green card through the EB3 process. The EB3 visa and green card are not available if you were not inspected and admitted into the U.S.
What if I entered without inspection, but then I received TPS after entry? You cannot adjust status via EB3 even if you were granted TPS after you entered the U.S. without inspection. Before 2021, some EB3 applicants who had TPS could adjust status depending on what state they lived in, even if they entered the U.S. without inspection. But in 2021, the U.S. Supreme Court ruled that this is no longer possible, no matter where a TPS holder lives. If a person originally enters the U.S. without inspection, the grant of TPS at a later date is not considered a lawful admission that makes a person eligible to adjust status.
How about if I have TPS, then travel on TPS advance parole and re-enter the United States lawfully? No, not if you are applying for an employment-based EB3 visa. If you last traveled and re-entered the U.S. on TPS advance parole prior to August 20, 2020, you might be able to adjust status through a family-based application as an immediate relative. But this is not available for employment-based EB3 visas, because you need to show you originally entered the United States lawfully, prior to receiving TPS, to qualify for an employment-based EB3 visa.