Combine the watermelon, mint leaves, simple syrup and lime juice in a large bowl.
Use a blender combine the mixture into a blender until smooth. Strain the purée into a bowl, collecting as much juice as possible without forcing any solids through the strainer.
Pour the juice into 8 mason jars. Add any additional lime juice to taste. Fill each with 1/2 cup water and gently stir.
Sprinkle the yeast onto the top of each jar and screw on the cap. Shake to dissolve and distribute the yeast. Let the bottles sit at room temperature out of direct sunlight until carbonated, typically 12 to 48 hours, depending on the temperature of the room.
Refrigerate the watermelon soda overnight or for up to 2 weeks. Open very slowly over a sink to release the pressure gradually and avoid bubble-ups.