Whisk together 2 tablespoons red wine vinegar, 1 tablespoon lemon juice, 1 teaspoon ground allspice, and salt and pepper to taste. Marinate 1 pound beef or lamb stir-fry strips in the mixture at least 2 hours and up to overnight, refrigerated.
Prepare tahini sauce by blending 1 cup tahini with 1/2 cup fresh lemon juice, 2 to 3 cloves garlic, the leaves of 1/2 bunch Italian parsley, and about 1 cup water, enough to make a thin, creamy consistency.
Prepare an onion relish by combining 1 diced sweet or red onion, 1/2 cup coarsely chopped Italian parsley, and 1 tablespoon ground sumac.
Drain the meat, then grill or broil it and serve in pita breads, topped with the tahini sauce and onion relish.