Fill a large, shallow, straight-sided pan with 2 inches water. Bring to a boil. Break each egg into a small ramekin.
Meanwhile, heat oil in a medium nonstick skillet over medium heat until hot but not smoking. Add tomatoes; cook, stirring, until they start to break down, about 5 minutes. Add salt; season with pepper. Transfer to a plate; tent with foil to keep warm. Add spinach and garlic to skillet; cook, stirring, until spinach has just wilted, 1 to 2 minutes. Discard garlic.
When water is boiling, add vinegar; turn off heat. Using tongs, tilt ramekins in pan, and slide eggs into water. Cover; let stand until whites are opaque and yolks are cooked as desired, 2 to 3 minutes. Use a slotted spoon to put eggs on paper towels to drain. Trim whites.
Divide spinach among four plates; lay a poached egg and 8 tomato halves on top. Season with pepper; sprinkle each serving with 3/4 teaspoon chives. Note: Partially cooked eggs should not be prepared for pregnant women, babies, young children, the elderly, or anyone whose health is compromised.