1. Syrup: In a medium saucepan, combine water and sugars. Cook over medium heat until boiling. Turn heat to low, stir, and simmer about 30 minutes. Stir occasionally.
2. Pudding: Place grated ginger in cheesecloth; tie with string. Combine milk, soy milk, and ginger tea bag in a large saucepan. Simmer over low heat for 15–20 minutes. Occasionally skim top. Remove ginger bag.
3. Stir in rice and syrup. Cook on high until mixture boils. Reduce heat to medium-low and simmer, uncovered, stirring often, until pudding thickens and rice is tender, about 45 minutes. Add heavy cream.
4. Spoon pudding into each of 10 cups, topping each with a strawberry slice and the ginger, if desired.