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Haider al-Abadi still faces opposition closer to home, where his Shi'ite party colleague Nuri al-Maliki has refused to step aside after eight years as premier that have alienated Iraq's once dominant Sunni minority and irked Washington and Tehran.
However, Shi'ite militia and army commanders long loyal to Maliki signalled their backing for the change, as did many people on the streets of Baghdad, eager for an end to fears of a further descent into sectarian and ethnic bloodletting.
Sunni neighbours Turkey and Saudi Arabia also welcomed Abadi's appointment.
A statement from Maliki's office said he met senior security officials and army and police commanders to urge them "not to interfere in the political crisis". At least 17 people were killed in two car bombings in Shi'ite areas of Baghdad - a kind of attack that has become increasingly routine in recent months.
As Western powers and international aid agencies considered further help for tens of thousands of people driven from their homes and under threat from the Sunni militants of the Islamic State near the Syrian border, Secretary of State John Kerry said the United States would consider requests for military and other assistance once Abadi forms a government to unite the country.
Underscoring the convergence of interest in Iraq that marks the normally hostile relationship between Washington and Iran, senior Iranian officials congratulated Abadi on his nomination, three months after a parliamentary election left Maliki's bloc as the biggest in the legislature. Like Western powers, Shi'ite Iran is alarmed by Sunni militants' hold in Syria and Iraq.
"Iran supports the legal process that has taken its course with respect to choosing Iraq's new prime minister," the representative of Supreme Leader Ayatollah Ali Khamenei on the Supreme National Security Council was quoted as saying.
"Iran favours a cohesive, integrated and secure Iraq," he said, adding an apparent appeal to Maliki to concede.
Abadi himself, long exiled in Britain, is seen as a far less polarising, sectarian figure than Maliki, who is also from the Shi'ite Islamic Dawa party. Abadi appears to have the blessing of Iraq's powerful Shi'ite clergy, a major force since U.S. troops toppled Sunni dictator Saddam Hussein in 2003.
Iraqi state television said Abadi "called on all political powers who believe in the constitution and democracy to unite efforts and close ranks to respond to Iraq's great challenges".
One politician close to Abadi told Reuters that the prime minister-designate had begun contacting leaders of major groups to sound them out on forming a new cabinet. The president said on Monday he hoped he would succeed within the next month.
A statement from a major Shi'ite militia group, Asaib Ahl Haq, which has backed Maliki and reinforced the Iraqi army as it fell back from the north in June, called for an end to the legalistic arguments of the kind used by Maliki to justify his retaining power and urged "self-restraint by all sides".
It said leaders should "give priority to the public interest over the private" and respect clerical guidance - a clear reference to indications that Grand Ayatollah Ali al-Sistani favours the removal of Maliki to address the national crisis.
While U.S. officials have been at pains not to appear to be imposing a new leadership on Iraq, three years after U.S. troops left the country, President Barack Obama was quick to welcome the appointment. Wrangling over a new government since Iraqis elected the new parliament in April has been exploited by the Islamic State to seize much of the north and west.
Obama has sent hundreds of U.S. military advisers and last week launched air strikes on the militants after they made dramatic gains against the Peshmerga forces of Iraq's autonomous ethnic Kurdish region, an ally of the Baghdad authorities.
Kurdish president Masoud Barzani told U.S. Vice President Joe Biden that he would work with Abadi, the White House said.
U.S. officials have said the Kurds are also receiving direct military aid, and U.S. and British aircraft have dropped food and other supplies to terrified civilians, including from the Yazidi religious minority, who have taken refuge in remote mountains. The United Nations said on Tuesday that 20,000 to 30,000 Yazidis may still be sheltering on the arid Mount Sinjar.
Kerry, who on Monday had warned Maliki not to resort to force to hold on to power, said on Tuesday that Abadi could win more U.S. military and economic assistance.
"We are prepared to consider additional political, economic and security options as Iraq's government starts to build a new government," he told a news conference in Australia, where he also reaffirmed that Washington would not send combat troops.
"The best thing for stability in Iraq is for an inclusive government to bring the disaffected parties to the table and work with them in order to make sure there is the kind of sharing of power and decision-making that people feel confident the government represents all of their interests," Kerry added.
It remains unclear how much support Maliki, who remains acting premier, has to obstruct the formation of a new administration. One senior government official told Reuters that his fears of a military standoff in the capital had eased as police and troops had reduced their presence on the streets.
"Yesterday Baghdad was very tense," he said. "But key military commanders have since contacted the president and said they would support him and not Maliki."
In both Shi'ite and Sunni districts of the capital, many spoke of a sense of relief and cautious hope for change.
"I'm very happy Maliki will not be prime minister again. I hate him; he killed my sons and broke my heart," said 68-year-old Um Aqeel as she walked in the Karrada shopping district.
Saying two of her sons had died in violence in the past year - one while serving as a soldier in the north in May - she said: "Maliki knows only the language of war and never believes in peace, just like Saddam. Yesterday when I heard he was out I felt justice has been done by God, and my two beloved sons who were killed because of him will rest in peace."
But as Um Aqeel offered sweets to passers-by in the mainly Shi'ite area to share her satisfaction, one man, Murtadha al-Waeli, warned her angrily that she was wrong to celebrate.
"Soon you will all regret Maliki's going," he said. "It was he who built a strong army. Iraq will fall apart after Maliki, and we will lose the battle with the terrorists. Shi'ites will pay a high price for losing Maliki. Just wait and see."
In the mainly Sunni district of Adhamiya, where many people have long resented what they saw as Maliki's determination to keep Sunnis out of positions of influence, cafe owner Khalid Saad said he hoped Abadi would learn a lesson from the past by keeping his distance from Iran and leaving Sunnis in peace.
"Maliki treated us Sunni like aliens," he said. "We hope Abadi will learn from Maliki's fatal mistakes and pull the country back from its sea of troubles." (Additional reporting by Raheem Salman in Baghdad and Lesley Wroughton in Sydney and; Writing by Alastair Macdonald; editing by David Stamp)