Drug Cartel Terrorist Designation Act This bill directs the Department of State to designate four specified drug cartels as foreign terrorist organizations. (Among other things, such a designation allows the Department of the Treasury to require financial institutions to block transactions involving the organization.) The four specified cartels in the bill are the Gulf Cartel, the Cartel Del Noreste, the Cartel de Sinaloa, and the Cartel de Jalisco Nueva Generacion. The bill also requires the State Department to submit a detailed report on those four cartels and any other cartels it may identify. Based on this report, the State Department must designate as a foreign terrorist organization any such identified cartel (or faction thereof) that meets certain criteria for designation as a foreign terrorist organization. The bill specifies that it may not be construed to expand eligibility for asylum.