Slice tempeh lengthwise into 8 (1/4-inch-thick) slices.
Heat oil in a large nonstick skillet over medium. Place tempeh slices in skillet, and cook until golden-brown, 3 to 4 minutes on each side.
Meanwhile, stir together 1/2 cup water, minced onion, soy sauce, rosemary, oregano, 2 minced garlic cloves, and 1/8 teaspoon black pepper in a small bowl.
Add soy sauce mixture to tempeh in pan. Cover, and reduce heat to medium-low. Braise tempeh, turning occasionally, until liquid has mostly evaporated and tempeh has absorbed the flavors, about 10 minutes.
Process silken tofu and remaining 2 garlic cloves in a high-powered blender until smooth. Place in a bowl. Stir in cucumber, lemon juice, dill, vinegar, salt, and remaining 1/8 teaspoon black pepper.
Divide tempeh evenly among flatbreads, and top each with about 2 tablespoons tzatziki. Divide onion slices and tomatoes evenly among gyros.