District of Columbia Prosecutor Home Rule Act of 2021 This bill shifts the responsibility for conducting prosecutions for violations of District of Columbia laws from the Corporation Counsel for the District of Columbia (Attorney General for the District) or his or her assistants to the head of the office designated under District local law as the one responsible for conducting such prosecutions (local prosecutor's office) or his or her assistants. An indictment or information brought in the name of the United States in the U.S. District Court for the District of Columbia may include charges of offenses prosecutable by the District if the head of the local prosecutor's office consents to such inclusion in writing, and the District in the Superior Court of the District of Columbia may be joined for trial in the U.S. District Court for the District of Columbia with an indictment or information brought in that court if the offenses charged could have been joined in the same indictment or information and if the head of the local prosecutor's office consents to such joinder.