Cut enough pita bread into ½-inch pieces to make ¼ cup (about one-eighth of 1 pita bread). Cut remaining pita breads into wedges for serving. Process pita pieces, tomatoes, and half of red peppers in blender until smooth, 1 to 2 minutes. Cut remaining red peppers into ¼-inch pieces and set aside.
Heat oil in 10-inch skillet over medium heat until shimmering. Add garlic and cook, stirring occasionally, until golden, about 1 minute. Add tomato paste, coriander, paprika, cumin, salt, pepper, and cayenne and cook, stirring constantly, until rust-colored and fragrant, about 1 minute. Stir in tomato–red pepper puree and reserved red peppers (mixture may sputter) and bring to simmer. Reduce heat to maintain simmer; cook, stirring occasionally, until slightly thickened (spatula will leave trail that slowly fills in behind it, but sauce will still slosh when skillet is shaken), 6 to 8 minutes.
Remove skillet from heat. Using back of spoon, make 4 shallow dime-size indentations in sauce. Crack 1 egg into small bowl and pour into 1 indentation (it will hold yolk in place but not fully contain egg). Repeat with remaining 3 eggs. Spoon sauce over edges of egg whites so that whites are partially covered and yolks are exposed.
Bring to simmer over medium heat (there should be small bubbles across entire surface). Reduce heat to maintain simmer. Cover and cook until yolks film over, 3 to 4 minutes. Continue to cook, covered, until whites are softly but uniformly set (if skillet is shaken lightly, each egg should jiggle as a single unit), 1 to 2 minutes longer. Off heat, sprinkle with cilantro, feta, and olives. Serve immediately, passing pita wedges separately.
Recipe from America's Test Kitchen.