Return the black-eyed peas to the pot. Pour in enough boiling water to cover them by 2 inches (5 cm). Simmer the black-eyed peas over low heat for about 35 minutes, or until tender but al dente. Drain immediately into a colander and rinse.
While the black-eyed peas are cooking, cut the leeks into ½-inch (1.5 cm) rounds. Rinse them very well to remove any grit. Spin dry in a salad spinner.
In a large skillet, heat 2 tablespoons of the olive oil over medium-high heat. Add the leeks, season lightly with salt, and cook, stirring occasionally, until the leeks begin to brown lightly.
In a separate skillet, heat 2 tablespoons of the olive oil over medium heat. Add the spinach and cook, stirring occasionally, until the leaves are wilted and the liquid they have released has cooked off. Add the garlic and cook until fragrant. Remove from the heat.
Combine the black-eyed peas, spinach-garlic mixture, leeks, and mint in a serving bowl. Add the remaining olive oil and remaining 1 tablespoon vinegar. Season to taste with salt and pepper, garnish with the pomegranate seeds, and serve.