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Heavy fighting between Yemeni pro-government forces and Houthi rebels has killed at least 25 people since Tuesday in the besieged southwestern province of Taiz, Yemeni security officials said on Thursday.

Houthis carried out an attack with heat-seeking, shoulder-fired missiles against pro-government forces led by the nephew of Yemen's late President Ali Abdullah Saleh in the western al-Burj district in Taiz, the officials said.

More than 10 fighters of Brig. Gen. Tareq Mohammed Abdullah Saleh's forces were killed in the attack and another eight were wounded, they said.

Loyalists to Ali Abdullah Saleh vowed revenge on the Houthis, with many of whom joining ranks with Tareq Saleh.

The Saudi-led coalition is reportedly supporting Tareq Saleh’s militias.

In a separate battle in southern Taiz, pro-government forces carried out a fierce attack against the rebels aimed at seizing the town of al-Rahda, military spokesman Ahmed el-Naqib said. The Saudi-led coalition is carrying out airstrikes against the rebels there.

The fighting killed more than 15 fighters from both sides in the past 24 hours, security officials said.

Forces loyal to Hadi also seized control of several villages in the Qabbaytah district in the southern province of Lahj after heavy clashes left five Houthi rebels dead, the officials said.

The security officials spoke on condition of anonymity because they weren't authorised to speak to the press.

Yemen has been embroiled in a war between the Houthi rebels and Abed Rabbo Mansour Hadi’s Riyadh-based government, which is allied with a Saudi-led military coalition. 

The coalition, which has been blasted for contributing to a high civilian death toll in its operation, says it aims to restore the government of self-exiled Hadi.

More than 10,000 people have since been killed in Yemen's conflict, in what the United Nations has called the world's worst humanitarian crisis.

"We have called on all parties to take appropriate measures to mitigate the risk of civilian casualties," Haley said. "The United States is working to offer support to the Saudi-led Coalition, while minimising civilian casualties."