By Jim Muir
BBC News, Baghdad
At a glittering reception at the British embassy in Iraq, the BBC's Jim Muir goes in search of the answer to a prickly question: Who will be Iraq's next prime minister?
A power struggle between Mr Allawi and Mr Maliki has stalled Iraq's new government
It's now well over three and a half months since general elections were held in Iraq, producing an inconclusive result.
The secular Sunni alliance headed by Iyad Allawi came out slightly ahead of the mainly Shia bloc of incumbent Prime Minster Nouri al-Maliki.
But neither can form a government alone, and putting together a coalition is proving difficult and time consuming.
Meanwhile, bombs are still going off, and the Americans are preparing to withdraw all their combat troops by the end of August, further adding to the jitters.
Even in Baghdad, our sovereign lady Queen Elizabeth celebrates her official birthday around this time of the year.
She doesn't turn up herself, of course.
It's really a sort of British national day, when the embassy opens its doors to the great and the good of Baghdad society and offers them a modestly glittering reception which, in keeping with the times, is sponsored by various banks and companies whose placards adorn the chandeliered ballroom.
So it offered a good opportunity to catch, in one room, a fair cross-section of the Iraqi and foreign political elite.
There was a sprinkling of prominent government ministers, mingling with political personalities and party officials, high-ranking military men, and a clutch of ambassadors of various nations.
As I flitted from one to another, I made a point of asking them all the same question: So who's going to be the next Iraqi prime minister?
Here we were, well over three months after general elections, and the amazing thing was, not a single person had a clear answer.
It wasn't as though they were trying to hide some secret to which they were privy.
They genuinely didn't know, because nobody does.
It's all the more amazing because there are only four substantial political blocs in the field, and an even smaller number of serious contestants for the prime minister's job.
A couple of days earlier, we'd been treated to another set-piece occasion: The first meeting of the new parliament, whose 325 members emerged successfully from the elections back on 7 March.
Actually they were one short - Bashar al-Ageidi, an elected MP for Mr Allawi's Iraqiyya bloc, had been shot dead outside his house up in Mosul.
But here were all the others, in their splendid diversity: Shia clerics in their white or black turbans, Arab tribesmen in their traditional headgear and robes, Kurdish leaders with their baggy pants and cummerbunds, and of course, plenty of others in smart western suits, all embracing and greeting one another as though they were the best of long-lost friends.
But the sheer multiplicity of the garb alone gave some hint of the sectarian and ethnic factionalism that bedevils Iraqi politics and makes it so hard for them to agree on anything.
Barrel of crabs
They were only meeting now because there was a constitutional deadline for the first session of parliament after the approval of the election results by the Supreme Court, which took nearly three months.
The MPs couldn't actually agree on anything, like electing their own Speaker, because that's part of the wider power-sharing package deal that's yet to be struck and which is certainly weeks away, possibly months.
So they just had to take the oath, declare the meeting open, and suspend it indefinitely.
To try to explain why what should be a fairly simple political situation is so deadlocked, is to enter a Byzantine world of convoluted rivalries complicated by shared interests on some levels and incompatible antagonisms on others.
To try to put it simply: Mr Allawi, a secular Shia backed by Sunnis, came out slightly ahead of the Shia religious bloc of Nouri al-Maliki, but neither has anything like enough seats to rule alone.
Now Mr Maliki has teamed up with the second-ranking religious Shia coalition to make one big alliance that far outnumbers Mr Allawi.
But Mr Maliki's problem is that his allies want his seats but they don't really want him back as prime minister.
Hence the endless round of probing and testing of possible alliances and permutations, a real barrel of crabs, and it's nowhere near done.
Many have expressed fears that, with the Americans now pulling out serious numbers of troops by the end of August, this political stagnation may create a dangerous vacuum which insurgents could exploit.
Iraq's banks have been targeted in recent attacks
It needn't necessarily be so. But they may have a point.
Just the day before that parliament meeting, the heavily-guarded Central Bank in the heart of Baghdad came under an extraordinary assault from gunmen wearing suicide vests.
They set off a bomb, got into the bank building and destroyed the contents of one particular room before blowing themselves up.
Although the Sunni extremist insurgent umbrella group said it carried out the attack, it left many questions, and strong rumours that it may have been aimed at covering up a huge money-laundering operation.
There was similar speculation after another suicide attack on the Trade Bank of Iraq a week later.
We may never know.
But as one of the survivors of that last bombing said: "This is the situation in the great Iraq... God help us."
It's all certainly a far cry from the simple vision of those who sent their troops in here in 2003, thinking that you just had to say words like freedom and democracy, and everything would be all right.
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