Stir together eggplant, 4 cups water, and ¼ cup salt in a large bowl. Place a small plate on top of eggplant to keep the cubes submerged; let soak 15 minutes. Drain eggplant well; pat and gently squeeze dry using a clean kitchen towel.

Heat 2 tablespoons oil in a large skillet over medium-high until shimmering. Add eggplant in a single layer; cook, stirring occasionally, until browned on all sides, about 7 minutes. Transfer eggplant to a medium bowl.

Add remaining 1 tablespoon oil to skillet, and reduce heat to medium. Add sweet peppers and onion; cook, stirring often, until beginning to soften, about 3 minutes. Add garlic, and cook, stirring often, until peppers and onion are tender and garlic is golden brown, 1 to 2 minutes.

Add harissa, paprika, sugar, cumin, and remaining 2 teaspoons salt to skillet. Cook, stirring constantly, until fragrant, about 45 seconds. Add eggplant and tomatoes with juices; cook, stirring often and mashing tomatoes into smaller pieces, until mixture is slightly thickened, about 5 minutes. Season with additional salt to taste.

Make 4 divots in tomato mixture using the back of a spoon; crack 1 egg into each divot. Cover and cook until egg whites are set but yolks are still jiggly, 3 to 6 minutes, or until desired doneness. Sprinkle with feta and additional paprika; garnish with cilantro. Serve with warm pita bread.