Place greens in a medium Dutch oven; add tomatoes. Cook over low heat until steam begins to form. Cover and cook 1 1/2 hours, stirring occasionally. Remove from heat, and cool slightly.
Strain cooled mixture through cheesecloth; discard turnip greens and tomato pulp. Clarify bouillon according to procedure.
Season bouillon with salt and pepper. Serve bouillon warm or chilled as an appetizer or use as a foundation for other soups or stews.