Combine 1⁄4 cup of water, soy sauce, sugar, garlic, lime juice, ginger, and black pepper in a saucepan. Bring to a boil, then reduce heat to low, and simmer, uncovered, for 2 minutes. Mix arrowroot powder with 1 tablespoon of water, then whisk mixture into sauce. Cook, stirring, until thickened and bubbly. Simmer for 1 minute more. Remove from heat.