In a small saucepan over low to medium heat, combine raspberries, strawberries, water and granulated sugar. Bring to a boil and stir until sugar dissolves and berries soften. Press the mixture through a fine-mesh strainer and discard the pulp and seeds. Let cool and add rum.
Pour the mixture evenly over crushed ice in serving glasses. Top with lemon-lime soda and garnish with fresh mint.