MP hopes Patagonian visit will help 'thaw' relations
Their sovereignty has been disputed for centuries. The islands have been invaded and fought over, leading to hundreds of deaths.
So how do you resolve the arguments between the UK and Argentina over the Falklands?
Enter David Davies and MPs on Parliament's Welsh affairs committee. Nine of its members will fly (business class, since you ask) to Buenos Aires on Sunday for what politicians like to call a fact-finding trip.
Their ultimate destination is Patagonia, where they will mark the 150th anniversary of the arrival of the first Welsh settlers. In their absence, Chancellor George Osborne will deliver his Budget and the Wales Office is expected to publish legislation to give the Welsh government control of some taxes for the first time. Ministers are also expected to give their response to the committee's recent report on the draft version of that legislation.
So why go? And why now? Committee chair Mr Davies told me: "We were invited by the Argentine government to come and visit. The Foreign Office were keen for us to go and visit because of, frankly, a lack of very friendly communication between the two countries for a number of years now. It's the 150th anniversary of the first Welsh landings in Patagonia.
"We are also in any case undertaking an inquiry into how Wales is marketed abroad and it would have been useful as part of that to go somewhere and see how that's done."
The £50,000 cost of the six-day visit (plus two days' travelling) is coming from a fund used for select committee travel. The Monmouth MP said: "The money couldn't have been used for anything other than select committee travel and I think there'll be many people in Wales who will be pleased that we recognise the Welsh language in Patagonia, which is still going strong after 150 years."
He said he hoped the visit would help celebrate the anniversary and build on links between the two countries, although no-one is expecting the future of the Falklands to be discussed.
"I hope that it leads to a slight thaw perhaps in the relationship between the UK and Argentina. They are potentially a huge trading partner for us and I hope we learn something about how Wales is marketed because I don't feel that's been done as well as Scotland or even some regions of England and if we can do anything that improves the way that Wales is marketed then that could be very good for our tourism industry."
Only three committee members - Glyn Davies, Geraint Davies and Karen Lumley - won't be making the trip. (I'm counting them all out and counting them all back.) I'm told that business class on Air France also costs less than flying economy on British Airways and that's the reason for the choice of airline, rather than any committee member's ideological objection to the word "British".
The MPs should return in time for the key House of Commons votes on the Budget statement they missed.