Heat oil in a saucepan over medium-low heat. Add anchovy fillets and cook, stirring often, until they begin to sizzle, about 1 minute. Stir in garlic and cook just until fragrant, about 1 minute. Add fresh oregano and reduce the heat to low; cook until oregano is wilted, 2 to 3 minutes.
Add tomatoes, pepper flakes, and dried oregano; bring to a simmer. Season with sugar, pepper, and salt; reduce the heat to low and simmer, stirring occasionally, until thickened and oil rises to the top, 35 to 40 minutes. Stir in baking soda until thoroughly incorporated.
