Combine turbinado sugar and gelatin together in a large saucepan; add coconut milk and rice milk. Cook and stir milk mixture over medium heat until almost boiling and sugar is dissolved, about 5 minutes.
Scoop about 1/2 cup hot milk mixture into a bowl with the beaten egg yolks; whisk together. Slowly pour egg mixture back into the saucepan, stirring constantly. Lower heat to medium-low to avoid curdling; cook and stir until mixture is smooth, about 3 minutes; do not boil. Remove from heat and cool slightly.
Place strawberries in a blender and pulse several times to chop before leaving it on to puree.
Stir strawberry puree and vanilla extract into milk mixture. Refrigerate until very cold, 8 to 48 hours. Pour chilled strawberry mixture into a 4-quart ice cream maker and follow manufacturer's instructions.
Place about 2 cups fresh strawberries in a blender to make about 1 cup strawberry puree.