Hundreds of thousands of Yemenis have demonstrated in nearly every major city of the country, demanding trial for the family and close aides of the ailing president.

Tuesday’s rallies were the largest protests since President Ali Abdullah Saleh went abroad for medical treatment for injuries suffered in an attack on his compound. Some of Mr. Saleh’s family and closest aides remained behind, and Yemen remains locked in a power struggle between the president’s allies and tribesmen demanding an end to the regime’s nearly 33-year rule.

In Washington, the State Department’s counter-terror coordinator said the U.S. is worried that the ongoing unrest in Yemen could fuel connections between al-Qaeda-linked militants there and al-Shabab insurgents in Somalia.

The American official, Daniel Benjamin, said insurgents in Yemen are trying to take advantage of the turmoil in their country, are operating more in the open and have been able to acquire and hold more territory.

Residents in Shabwa, one of the al-Qaeda strongholds in southern Yemen, are reporting intensifying overflights by U.S. drones, suggesting the Americans are keeping close watch on the situation.

The CIA is trying to speed up construction of a Persian Gulf base for its drones, but the process is being held up by logistic delays, U.S. officials said. The officials, speaking on condition of anonymity to discuss sensitive security matters, said the base is at least eight months away from completion.

The Associated Press has withheld the exact location at the request of U.S. officials.

In Riyadh, Saudi Arabia, foreign ministers of the Gulf Cooperation Council discussed ways to resurrect their initiative aimed at resolving the crisis. The United States and Saudi Arabia are trying to persuade the ruling party to move forward with the GCC deal that would end Mr. Saleh’s rule, hand power to his vice president, create a unity government and conduct elections within two months. Mr. Saleh accepted the idea three separate times but balked at the last minute.

On Tuesday, tens of thousands of young people crowded outside the home of acting president Abed Rabbo Mansour Hadi, who assumed power when Mr. Saleh left. The crowd vowed to stay outside Mr. Hadi’s home, which is protected by special forces led by Mr. Saleh’s son, until their demands were met.

Many in Mr. Saleh’s inner circle remain in positions of power, including his son Ahmed, who commands the special forces and Yemen’s Republican Guard. Mr. Hadi’s role in the power struggle is unclear, but he has met with the opposition, suggesting he’s willing to exercise some constitutional authority.

Mr. Saleh is being treated in the Saudi capital, Riyadh, along with several government officials wounded in the June 3 rocket attack on the presidential compound.

A senior Yemeni official in Riyadh said Mr. Saleh had developed a problem with his throat. He did not elaborate. The official asked not to be named because of the sensitive nature of the information.

Some in the crowd of young people said Mr. Hadi “shoulders the legal and ethical responsibility of any crimes that might be committed by Saleh’s sons or the regime’s remnants.” They said they would not negotiate with anyone in the regime.

There were similar demonstrations in Hadramawt, Hodeida, Ibb, Damar and Saada.

In Taiz, Yemen’s second-largest city and the site of major protests, demonstrators clashed with Republican Guard units guarding the city’s presidential palace and several ambulances were seen rushing to the area. There were no official casualty reports.

The Russian government evacuated its nationals, an airport official and a government official said on condition of anonymity because they were not authorized to speak to the media.

The United Nations secretary general’s envoy, Gamal bin Omar, arrived in Sanaa on Tuesday and is meeting with the acting president Hadi. This is Mr. bin Omar’s third visit since the crisis broke out in Yemen in February.

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