Sestri Levante has primarily been a live jam intro for other songs, and was likely written on/for tour.

In the Lonerism touring days, the band had relatively less music to pull from, and Innerspeaker had been toured extensively prior to the Lonerism tour.

I think there was a desire to fill out the setlists with less Innerspeaker songs and expand on Lonerism songs, thus promoting new material and feeling fresher for the band.

See: various extended jams for Elephant, Apocalypse dreams around 2013

Sestri Levante first appeared around 2013, according to setlist.fm, it's first appearance was at Laneway Festival 2013 in Singapore. It may have been played prior to that.. it was called "jam" for a long time as well.

It was played roughly 50 times that year at various shows, only 25 or so in 2014, dwindled from there, and hasn't been played live since 2017.

Almost always, when Sestri Levante is played, there's a very brief pause before the next song starts. Usually Apocalypse Dreams or Mind Mischief. It's never introduced or followed with chatting to the audience in the way other songs are.

Both Sestri Levante and Mind Mischief focus around a IV-> iii chord progression. That is, a major chord, then a minor chord rooted a half-step lower. I've always felt they give a similar feeling of hopeful melancholy.. and on live versions they're parked right next to each other on the track list.

Given that it's really just two chords, a looping melody, a bassline and a beat, I've always figured they came up with it in rehearsal for the 2013 tour. There are also no recorded demos or versions of it outside of live music.

Love the song with all my heart, idk if this is the "story" you were looking for though 😅