By Yomna Ehab and Parisa HafeziCAIRO/DUBAI, Aug 27 (Reuters) - Qatar's prime minister visited Tehran on Thursday in a bid to relaunch diplomacy after the U.S. and Iran traded recriminations over Washington's promise to increase economic pressure, a move one Iranian official described as "all-out economic war".A Gulf neighbour of Iran and a U.S. ally, Qatar has served as a back-channel negotiator for the warring parties and played a direct role in securing a June ceasefire that briefly led to a cessation of hostilities.Qatar's Prime Minister Sheikh Mohammed bin Abdulrahman Al Thani, who is also his country's foreign minister, met Iranian Foreign Minister Abbas Araqchi, according to Iran's foreign ministry channel on Telegram. The visit comes as the war nears its sixth month with fighting largely paused but no diplomatic breakthrough in sight. Both sides are at odds over control of the Strait of Hormuz, a chokepoint for global oil supplies that Tehran has used as leverage.A memorandum of understanding reached between the U.S. and Iran in June to end the conflict broke down in part because of differences over the narrow waterway between Iran and Oman where one-fifth of the world's oil supplies passed before the war began on February 28.Qatar and Iran discussed a proposed phased framework that includes establishing a temporary joint Iranian-Omani shipping corridor through the strait and carrying out a joint project to clear mines from the waterway, Qatar's foreign ministry said. The two officials also reviewed the latest developments in the region and efforts to de-escalate tensions and create conditions for dialogue, the ministry said. 'ALL-OUT ECONOMIC WAR'Although the strikes have stopped, Washington is now piling on economic pressure by announcing plans to cut Iran off from its trading partners with increased sanctions.Iranian Parliament Speaker Mohammad Baqer Qalibaf on Thursday called for national unity in the face of what he described as "all-out economic war", warning that internal division was "exactly what the enemy wants".Ebrahim Azizi, the head of the Iranian parliament's national security committee, said the sanctions were an "inhumane and hostile act" which nonetheless had lost their effectiveness. Tehran has lived under various forms of U.S. sanctions since the Islamic revolution in 1979.Hormuz oil flows have fallen to a three-month low, with just 5 million barrels per day transiting on Monday, according to the latest ship-tracking data. Before the war, the strait carried roughly 20 million barrels per day of crude oil.Prices rose again on Thursday, with Brent crude futures up 0.6% at $88.35 a barrel, as investors weighed the prospects for reopening of the strait.CONFLICTING ACCOUNTS OVER OMAN-IRAN TALKSIran's Revolutionary Guards said on Wednesday that Iran and Oman had agreed on how to share the waterway and its revenues, but a senior Iranian source later said the two countries were still working on the details of an agreement.Despite an aerial bombardment earlier in the war that severely diminished Iran's conventional forces, Tehran has maintained enough missile and drone capability to attack Gulf neighbours that house U.S. military bases and disrupt oil tankers in the strait.The International Maritime Organization has reported 70 incidents in the Strait of Hormuz since February 28 that have killed 19 seafarers.Thousands of others have died, mostly in Iran and Lebanon, since the U.S. and Israel attacked Iran with the stated goals of denying Iran a nuclear weapon, helping Iranians topple their government, and destroying Tehran's missile programme.An unidentified projectile hit a tanker in the strait on Tuesday to set off a fire that was later doused, but all crew were safe and accounted for, the United Kingdom Maritime Trade Operations agency said.The Strait of Hormuz was open to free navigation before the war, but Iran has since clung to its control of the waterway as leverage and a revenue source, seeking to charge ships a commission, which the U.S. has opposed."An agreement with Oman over the Strait of Hormuz has not been finalised," the senior Iranian source told Reuters on Wednesday, adding that talks on its details were continuing.That seemed to contradict a statement by Brigadier General Hossein Mohebbi, a spokesperson for the Islamic Revolutionary Guard Corps, who said Iran and Oman had "reached results that are acceptable to both sides" on controlling the strait and its revenues.Mohebbi said the strait would remain shut unless the U.S. met Tehran's conditions under the interim ceasefire that the two sides signed in June before it unravelled. Those conditions include an end to the U.S. blockade on Iranian ports, compensation and removal of sanctions. (Reporting by Yomna Ehab and Ahmed Tolba in Cairo and Parisa Hafezi in Dubai; Writing by Daniel Trotta, Stephen Coates, Conor Humphries and Sharon Singleton; Editing by Sanjeev Miglani, Clarence Fernandez and Gareth Jones)