Place tomatoes in a large bowl; add equal amounts balsamic vinegar and rice vinegar until tomatoes are covered. Season with basil and ginger; stir. Let sit, stirring every 15, until flavors, combine, about 1 hour.
Drain off any excess liquid from the tomatoes, reserving the vinegar mixture. Do not rinse the tomatoes.
Combine tomatoes and chayote and serve with vinegar mixture as the dressing.