Real Courts, Rule of Law Act of 2022 This bill establishes the U.S. Immigration Courts, a system of courts to be independent of the executive branch. The bill also transfers all functions of the Executive Office for Immigration Review (EOIR) to the Immigration Courts, with some exceptions. (Currently, the EOIR, located in the Department of Justice, has authority to adjudicate immigration cases.) The trial division of the Immigration Courts shall have jurisdiction over various immigration-related matters, including (1) removal proceedings, (2) reviews of rescissions of lawful permanent resident status, and (3) reviews of credible fear determinations in applications for refugee status. The appellate division shall have jurisdiction over appeals of decisions by the trial division and certain Department of Homeland Security determinations. The bill establishes various requirements and authorities relating to the Immigration Courts, including qualifications, length of term in office, and mandatory retirement age for judges. Each appellate division judge must be appointed by the President with the advice and consent of the Senate, and the appellate division shall appoint the trial division judges.