Real Justice for Our Veterans Act of 2021 This bill expands data collection on the effectiveness of veteran treatment court programs, incentivizes local governments to improve retention rates in veteran treatment court programs and drug court programs, and expands options for veterans to participate in drug court programs. First, the bill requires the Department of Justice (DOJ) to report on the effectiveness of veteran treatment court programs. The report must assess the population served by the programs, whether the programs use evidence-based treatments, the recidivism rates of participants, and program completion rates. The report must also assess whether women and racial and ethnic minorities have equal access to the programs and an equal opportunity to participate. Second, the bill requires DOJ's Bureau of Justice Assistance to implement a pilot program to make grants for local governments to improve retention in veteran treatment court programs and drug court programs. Finally, this bill allows a veteran (including a veteran who is a violent offender) to participate in a drug court program instead of a veteran treatment court program if a jurisdiction does not operate a veteran treatment court program.