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On October 25, 2011, ANTI- announced Islands' fourth record, A Sleep & a Forgetting, which was released February 14, 2012. In a statement, Thorburn explained, "I left New York after the end of a relationship and came to Los Angeles. There was a piano where I was staying and that’s where I wrote these songs. This record deals with loss, with memory and forgetting and with dreaming. I started writing it on Valentine’s Day and it’s coming out on Valentine’s Day." Drums on the record were played by Luc Laurent, who officially joined the band in November 2011 after his band Pepper Rabbit broke up. The song "Hallways" from the album was featured in a commercial for Dell and the Microsoft Surface tablet.
During the summer of 2015, Islands simultaneously recorded two records: Should I Remain Here, At Sea? — a spiritual successor to the band's debut album, Return to the Sea — and Taste, a "more electronic" album "buoyed by drum machines, programming and vintage synths." The albums were released on May 13, 2016 via the band's own Manqué label, reaching #21 and #23 on the Billboard Heatseeker charts, respectively.