How to Make It
In a medium saucepan, combine the water with 1 cup of the lemon juice and 1 cup of the sugar and bring to a boil, stirring to dissolve the sugar. Add the tapioca and cook over moderate heat, stirring often, until the pearls are plumped and tender, 3 to 5 minutes. Remove the tapioca from the heat and let cool to room temperature, stirring occasionally. Do not refrigerate.
Meanwhile, in another medium saucepan, combine the diced peaches, blueberries, smashed ginger and salt with the remaining 1 1/2 tablespoons of lemon juice and 1/4 cup of sugar and bring to a boil. Simmer over moderately low heat until the compote is deep purple and slightly thickened, about 10 minutes. Scrape the compote into a heatproof bowl and let cool completely. Pick out the ginger.
Scoop vanilla ice cream into glasses or bowls and top with the peach-blueberry compote. Garnish the sundaes with the tapioca and serve.