In a small saucepan over low heat, melt the butter and sugar together until the sugar is fully dissolved.
Stir in the tomato paste and cook gently, stirring, until smooth and fully incorporated into the butter mixture.
Remove from heat. Add 1/2 cup cold water and whisk until smooth.
Add the paprika, garlic powder, and ground ginger. Stir well to combine.
Mix in the soy sauce, rice vinegar, and sesame oil. Allow the mixture to cool completely to room temperature.
In a large bowl, add the Duke's mayonnaise. Pour in the cooled mixture and whisk until completely smooth and well combined.
Adjust the consistency with additional water, adding a splash at a time, until it reaches your preferred thickness. Thicker for dipping, thinner for drizzling over rice or using as a dressing.
Refrigerate for at least 1 hour before serving. The flavor improves significantly as it chills.