Make Syrup: Combine the water, sugar, corn syrup, and grated ginger in a saucepan over medium heat. Bring to a boil, stirring until the sugar completely dissolves.
Infuse: Remove from the heat, cover the pan, and let the ginger steep in the syrup for 20 minutes to extract its fiery flavor.
Chill: Strain the syrup through a fine-mesh sieve, pressing hard on the ginger solids to extract all the juices. Stir in the fresh lime juice, then chill the mixture in the refrigerator until completely cold (at least 4 hours).
Churn: Pour into your ice cream maker and churn according to the manufacturer's instructions. Freeze for 2 hours to firm up before serving.