Combine carrots, lime juice, vinegar, ginger, ketchup, maple syrup, water, tamari (or soy sauce) and salt in a blender. Pulse until finely chopped. With the motor running, slowly drizzle in oil and blend until smooth.
Place greens and cabbage in a large bowl. Add the dressing and toss.
Refrigerate dressing (Step 1) for up to 5 days.
People with celiac disease or gluten sensitivity should use soy sauces that are labeled "gluten-free," as soy sauce may contain wheat or other gluten-containing ingredients.