"(I realise there do have to be time limits, or nothing ever gets agreed.)"

The Papal Conclave system strictly speaking hasn't got a time limit. They just lock up all the cardinals together, and noone comes out till they've finished the job. I believe there is a fall back procedure for cutting off the food if they take too long, but it hasn't happened recently.

Of course for that you've got to have people with authority to make agreements, so you'd have to lock the heads of governments in. Though really the only one you'd really need would be the President of the USA, assuming you've got one.

It occurred to me how ironic it is in the light of the present shenanigans there - the first State in the Union due to be submerged is Florida isn't it?