Place garlic, ginger, tamarind sauce, rice vinegar and tea leaves in a container with a tight fitting lid. Add tofu cubes and toss well. Marinate all day or overnight, shaking the container occasionally to distribute the marinade.
Heat the oil in a large skillet over medium-high heat. Drain the tofu and reserve the marinade. Add tofu and tamari to taste to hot oil and cook, stirring almost constantly for 5 to 7 minutes, or until tofu is very hot. Transfer to a bowl and set aside.
Add additional oil to the hot skillet if needed, followed by the vegetables and remaining marinade. Again, season to taste with tamari. Cook for 5 to 7 minutes, stirring frequently, or until broccoli is crisp-tender and pan juices have evaporated. Return tofu to skillet and toss to combine. Drizzle with sesame oil and serve.