The phrase "be at cross-purposes" refers to a situation where two people or groups misunderstand each other's intentions or goals, causing conflict or confusion. Synonyms for this phrase include "talk past each other," "clash," "be at odds," "miscommunicate," or "have different agendas." In these situations, it is important for both parties to actively listen, clarify their intentions, and find common ground to resolve the conflict. Using clear and concise language, avoiding assumptions, and acknowledging each other's perspectives can help prevent misunderstandings and ensure effective communication.