Brazil’s tourist draws are legendary. There’s the Amazon, a wild expanse almost twice the size of India that’s can boast 40,000 plant species, 1,300 birds, the world’s longest river – and an opera house. Or what about Rio, with its riotous carnival, picturesque setting and iconic statue? Few waterfalls anywhere on the planet can trump Iguaçu for beauty.

For something more offbeat, it has Fernando de Noronha, whose beaches are regularly named among the world’s best, Sao Paulo, a booming hub known for its fashion, gastronomy and great hotels, and the Pantanal wetlands, home to jaguars and giant rodents.   

So why does it attract so few overseas visitors? According to a new World Travel and Tourism...