IATSE Projectionists Local 110 in Chicago was once widely believed to be one of the most mobbed up unions in America — a crooked outfit packed with members of the Chicago mob and their relatives, some of whom who are now drawing generous pensions from the local’s endangered pension plan.

Today, the local still counts relatives of long-dead gangsters among its members including the nephews of Sam Giancana, who ran Chicago’s organized crime syndicate in the late-1950s and ’60s and who was murdered gangland-style in 1975, and Joey “Doves” Aiuppa, the gangster who later succeeded Giancana and who many believe was behind his killing. According to financial statements filed with the U.S. Department of Labor, Giancana’s nephew Andrew sits on the local’s executive board, a post he’s held since 2004, while seven of the local’s members and prospective members — including the local’s secretary-treasurer — were convicted on...