Bloom the tomato paste. In a small saucepan, off the heat, whisk the tomato paste into the mirin until completely smooth. Skipping this leaves clumps that no amount of stirring will fix later.
[3 tablespoons tomato paste, 1/3 cup mirin]
Combine. Add the soy sauce, cane sugar, rice vinegar, ginger, green onion, garlic, sea salt, and sesame oil. Whisk together. Do not add water.
[1 cup soy sauce, 3/4 cup cane sugar, 2 tablespoons rice vinegar, 2 tablespoons fresh ginger, 3 tablespoons green onion, 1 tablespoon garlic, 1/4 teaspoon sea salt, 2 teaspoons toasted sesame oil]
Warm gently. Set over the lowest heat and stir until the sugar dissolves completely and the sauce turns glossy instead of grainy — 2 to 4 minutes. It should feel warm to the finger, around 120–140°F. Do not let it simmer or boil. Bubbles at the edge of the pan mean pull it off.
Cool and store. Let it come to room temperature, then pour into a clean glass jar. Do not strain — the green onion suspended in the sauce is part of what you're copying. It thickens slightly as it cools.