Heat olive oil over medium-low heat in a saucepan; stir anchovy fillets into olive oil and cook, stirring often, until the fillets begin to sizzle, about 1 minute. Mix garlic into oil and cook just until fragrant, 1 minute more. Add fresh oregano and reduce heat to low; cook until oregano is wilted, 2 or 3 more minutes.
Mix red pepper flakes, dried oregano, and tomatoes into olive oil mixture. Bring sauce to a simmer and season with salt, sugar, and black pepper. Turn heat to low; simmer sauce until thickened and oil rises to the top, 35 to 40 minutes, stirring occasionally.
Stir baking soda into pizza sauce, mixing until thoroughly combined.