Remove the pineapple skin and core completely. Cut the flesh into 1-inch chunks, removing any remaining eyes or tough spots.
Peel the ginger root using a spoon edge to scrape away the skin, then roughly chop into small pieces.
Add pineapple chunks, chopped ginger, and cold water to your high-speed blender or food processor.
Blend on high speed for 60-90 seconds until the mixture is completely smooth and no chunks remain.
Place a fine mesh strainer over a large mixing bowl and pour the blended mixture through it. Use a spoon to press the pulp and extract maximum juice.
Stir in fresh lime juice and taste. Add honey gradually if additional sweetness is desired, stirring until completely dissolved.
If using mint leaves, gently muddle 2-3 leaves in each serving glass before pouring the juice, or blend a few leaves with the pineapple for an infused flavor.
Serve immediately over ice, or chill in refrigerator for 30 minutes for best flavor. Stir before serving as natural separation may occur.