Preventing Sexual Harassment in Public Housing Act of 2020 This bill requires the Department of Housing and Urban Development (HUD) and the Department of Justice (DOJ) to take actions related to reporting and investigating housing-related discrimination and harassment, particularly sexual harassment. Specifically, HUD must include in its annual report to Congress tabulations of complaints of discriminatory housing practices filed with HUD or a fair housing assistance program in the preceding year. These tabulations must identify whether a complaint alleged discrimination based on race, color, religion, national origin, sex, handicap, or familial status. HUD must consider a complaint of sexual harassment under the category of discrimination based on sex and must include a separate tabulation of the total number of sexual harassment complaints filed. Additionally, DOJ must establish an initiative to investigate and prosecute allegations of housing-related sexual harassment.