Protect Americans from Foreign Criminals Act of 2021 This bill requires the Department of Homeland Security (DHS) to conduct background checks before releasing from custody an alien who is unlawfully present in the United States. Specifically, before releasing such an individual from detention while removal proceedings are pending, DHS must conduct a background check using criminal history databases covering the United States, the individual's country of origin, and any country that the individual transited through. DHS may not release the individual if the individual is (1) facing charges for (or convicted of) an aggravated felony, (2) a member of a criminal street gang, or (3) subject to mandatory detention on various crime- or security-related grounds. Furthermore, DHS must review and determine whether the criminal history databases of countries around the world meet standards for accessibility, accuracy, and other factors. DHS may not release a detained individual if a background check for that individual is required from a country with databases that fail to meet the relevant standards.