Stir together the ketchup, mayonnaise, ginger, sesame oil, soy sauce, vinegar and 1 tablespoon water in a small bowl. Set aside.
Heat the vegetable oil in a large skillet over medium-high heat. Add the garlic and cook for 30 seconds. Add the cremini and shiitake mushrooms and 2 tablespoons water. Cook, stirring, until the mushrooms are tender and golden, 7 to 8 minutes. Add the ketchup mixture and stir to coat. Transfer to a serving dish and top with chopped fresh cilantro.