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Stir 1 cup of water and the sugar in a small heavy saucepan over medium-low heat until the sugar dissolves.
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Slowly bring the syrup to a boil, then remove from the heat and cool the syrup completely.
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Spoon the passion fruit flesh into the cooled syrup and add the lemon juice.
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Pour the passion fruit mixture into an 8-inch square baking dish and freeze about 30 minutes, or until the mixture is icy around the edges.
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Using a large fork, stir the icy parts of the passion fruit mixture into the remaining mixture in the dish.
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Continue to freeze about 1 1/2 hours longer, or until the mixture is frozen, stirring the edges into the center every 20 to 30 minutes.
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Using a large fork, scrape the granita into flaky crystals. Cover and keep frozen.
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Gently toss the pineapple, papaya, mango, star fruit, and chopped mint in a large bowl.
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Spoon the mixed fruit into serving bowls and top with a large spoonful of granita.
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Garnish with mint sprigs and serve immediately.