To make grits, combine milk with water, butter, and salt in a large saucepan over medium-high heat. Bring to a roaring boil and slowly whisk in the grits. Decrease the heat to medium low. Continue whisking until no lumps remain, about 12 minutes. Decrease the heat to low. Cover the pan and let the grits simmer while preparing okra and tomatoes.
Heat olive oil in a medium to large skillet over medium-high heat. Add garlic and cook until golden.
Add okra and onion and season with salt and pepper to taste. Cook until okra is tender and bright, about 10 minutes.
Add cherry tomatoes and cook until bursting, about 4 minutes. Add vinegar and swirl together to coat.
To serve, remove grits from heat and stir in cheese. Divide grits among four bowls and spoon the okra mixture and pan sauce on top. Serve hot.