Combine pitted cherries, dried cherries, vinegar, water, garlic, sugar, ginger, allspice, cardamom, cinnamon and cayenne in a large saucepan. Bring to a simmer over medium-high heat. Reduce heat to medium-low and simmer, uncovered, stirring occasionally, until the fruit is tender, 15 to 20 minutes. Let cool slightly. Transfer to a blender. Cover the lid with a kitchen towel. Holding lid securely in place, blend until smooth. (Use caution when blending hot liquids.) Transfer the ketchup to a small bowl.
Make Ahead Tip: Cover and refrigerate for up to 2 weeks.