Heat a grill or grill pan to medium-high heat.
Grill the tomatoes skin-side down until dark grill lines appear, about 5 minutes, and then flip and cook until nicely charred and sagging, about 10 more minutes. Remove, chop into pieces, and transfer to a medium saute pan over medium heat. Stir in the vinegar, sugar, salt, aji amarillo paste, allspice, bay leaf and cinnamon stick and cook, stirring occasionally, until the mixture looks very dry and the tomatoes no longer hold their shape, 25 minutes.
Discard the cinnamon stick and bay leaf, and then blend the mixture to a smooth, thick sauce about the consistency of store-bought ketchup. If the ketchup is too thick, add 1 tablespoon water at a time and blend until desired consistency is reached.