The band was coming off the wildly huge success of Rio worldwide and making a lot of money. They realized however that this was not going to last forever. Thatcher's government still hadn't completely reformed the British tax code, so they decided to make the most of the opportunity by becoming tax exiles, spending most of the next year overseas recording the album (They also wanted a change of scenery, something John Taylor says was really, for him, a way of avoiding coming to grips with his growing drug dependency).
The first stop was the south of France. But it didn't go smoothly. The studio they used was tiny. They had to play in an upstairs booth, and check to see that it was recording properly downstairs. Oh, and there were no outlets down there so everything had to be run off a truck, with extension cords running all the way out to it. When equipment broke down, replacements had to be shipped from London, slowing things down.
That didn't stop them from laying down a few tracks, but it caused enough frustation and delays that they were as likely to be around the pool or cruising the restaurants and cafés in Cannes as recording. Plus, Taylor says, "I can state without fear of contradiction that this was Duran Duran's smoke period." He doesn't recall a single moment during this time when at least one member of the band wasn't either smoking a joint or rolling one. This led to the kind of long, deep, philosophical discussions you'd expect about things like the snare-drum part or the bass line.
They realized they needed another change of scenery and decamped to Montserrat, because they'd been heard the Police were having a great time there finishing up Synchronicity. They did indeed get more done, but Andy Taylor kept complaining about the still-inadequate equipment. Bills for their time at AIR Studios went unpaid to the point that they received a letter from the main office in London saying that they would not be able to use the facilities there until they paid up. They began to worry they wouldn't be able to make their deadline for the album.
In the middle of this they returned to Britain for the 1983 Prince's Trust benefit and another one in their hometown of Birmingham. The first one, performed before Charles and Di (they were her favorite band), was not a good performance (John Taylor says he couldn't keep his bass in tune, for one thing), largely due to all the stress they'd been were under, but it could have been worse—years later they and the world learned that an IRA bomb plot that night had been foiled by an informant. The latter show, at Aston Villa's home field, Villa Park, went a lot better.
Due to the problems in Montserrat, they decided to finish the album in Australia because they'd all enjoyed it very much when they'd been on tour there the year before. They did get it done, and on time, but the bad taste from the experience was such that John and Andy Taylor seriously considered quitting the band afterwards. Instead, they decided to put together the project that became the Power Station after the album and tour. But the rifts that developed in the band (Simon Le Bon and Nick Rhodes fronted a side project of their own, Arcadia, during that time) led to their ultimate departure from the band after A View to A Kill.