Heat light olive oil in a large saucepan over medium. Add onion and garlic, and cook stirring often; just until tender, about 5 minutes. Stir in turmeric, salt, pepper, nutmeg, and allspice; cook, stirring often, for 2 minutes. Add potato and stock, and bring to a simmer. Cook until potato is tender, about 10 minutes.
Reserve a handful of arugula for garnish. Add spinach, cilantro, and remaining arugula to the pan. Bring to a simmer over medium, and cook for 10 minutes. Working in batches, transfer the soup to a blender.
Secure the blender lid, and remove the centerpiece of the lid to allow steam to escape. Place a clean kitchen towel over the opening. Process until smooth. Taste and adjust seasonings if needed.
Divide the soup among 4 bowls.
Top each serving with a generous spoonful of Greek yogurt, some of the reserved arugula, and a drizzle of extra-virgin olive oil.