Near the town center are two beaches known locally as the Buccaneer Public Beaches; they're adjacent, appropriately enough, to the Buccaneer Club, on the island's sheltered western edge, facing Exuma Sound. Snorkeling is good here -- it's best where the pale turquoise waters near the coast deepen to a dark blue. Underwater rocks shelter lots of marine flora and fauna. The waves at these beaches are relatively calm.
On Eleuthera's Atlantic (eastern) side, about 1km ( 2/3 mile) from Governor's Harbour, is a much longer stretch of mostly pale pink sand, similar to what you'll find in Harbour Island. Known locally as French Leave Beach (or, less frequently, as the Club Med Public Beach), it's good for bodysurfing and, on days when storms are surging in the Atlantic, even conventional surfing.
Don't expect touristy kiosks selling drinks, snacks, or souvenirs at any of these beaches -- everything here is pristine and undeveloped.
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