Temporary Protected Status Reform and Integrity Act This bill imposes additional requirements and restrictions relating to temporary protected status (TPS). (Generally, a country may be designated for TPS for a period of time due to certain conditions in that country, such as civil unrest. Eligible nationals of that country may receive work authorization and may not be removed from the United States.) Under this bill, the Department of Homeland Security (DHS) may extend a TPS designation once, but an act of Congress shall be required for subsequent extensions. Currently, congressional action is not required for extensions. Furthermore, DHS may not designate a country for TPS if that country's designation terminated in the last 18 months under certain conditions. Specifically, this prohibition applies if the earlier designation terminated because (1) DHS did not make a determination that the conditions justifying the initial designation continue to exist, or (2) DHS recommended granting a subsequent extension but Congress did not enact a law to authorize that extension. Similarly, if a country's TPS designation terminated under these conditions, DHS may not designate that country again based on the justification that supported the original designation. The bill also establishes additional restrictions on individuals seeking or receiving TPS protections, such as prohibiting an individual who entered the United States unlawfully from receiving TPS, further limiting the authority of DHS to waive certain grounds of inadmissibility for a TPS applicant, and imposing additional restrictions on TPS recipients traveling abroad for brief trips (i.e., an individual who travels under certain conditions loses TPS).