The statement by Saudi Arabia comes after the separatists' Southern Transitional Council again claimed sole control on Sunday of Aden, a Red Sea port that serves as the seat of the internationally recognized government as Yemen's Shiite rebels, known as Houthis, hold the country's capital, Sanaa.
The council's decision adds yet more complexity to the grinding war in Yemen that has pushed the Arab world's poorest nation to the brink of famine and killed over 100,000 people.
The Saudi statement urged the council to return to the terms of the November 2019 Riyadh agreement, which ended earlier fighting between the separatists and the government forces — ostensible allies in the war against the Houthis — after the council seized control of Aden just a few months earlier.
That deal had called for both sides to remove heavy military equipment from Yemeni cities under their control and form a unity government that included equal representation. But that deal had yet to be implemented as the war continued, massive floods struck Aden and Yemen faces the threat of the coronavirus pandemic and the COVID-19 illness it causes.
In its statement, the Saudi-led coalition urged the council to honor the deal "to unite Yemeni ranks, restore state institutions and combat the scourge of terrorism."
The deal "entails forming a competent government" that would operate out of Aden "to tackle economic and developmental challenges, in light of natural disasters such as floods, fears of the COVID-19 pandemic outbreak and work to provide services to the brotherly people of Yemen."
The statement did not say what Saudi Arabia would do if the council refused. The kingdom, itself focused on the coronavirus outbreak, declared a unilateral cease-fire with the Houthis in April that was later extended through the holy Muslim month of Ramadan. But fighting has continued, with each side blaming the other for violations.
The Southern Transitional Council had been the on-the-ground allies of the United Arab Emirates, once Saudi Arabia's main partner in the war that subsequently withdrew from the conflict. The council's backers often fly the flag of former Communist South Yemen and have pushed to again split the country into two like it was from 1967 to 1990.
The Houthis in 2014 overran major parts of northern Yemen, including Sanaa, pushing out the internationally recognized government. The Saudi-led coalition entered the conflict backing the internationally recognized government in 2015.
In a separate development, Yemeni tribal leaders and security officials said Monday that two suspected U.S. drones strikes killed at least six al-Qaida militants in Yemen's southern province of Shabwa earlier this week. The U.S. military did not confirm the strikes.
The militants were killed in two separate strikes in the district of Bayhan, which is known as a stronghold of Yemen's al-Qaida branch, the security officials said. The officials spoke on condition of anonymity because they were not authorized to speak to reporters, while the tribal leaders did so for fear of reprisals.