In a large heavy pot, stir together canned tomato puree, onion, garlic, water, granulated sugar, distilled white vinegar, fine sea salt, ground black pepper, mustard powder, ground allspice, ground cinnamon, ground cloves, celery seed, and cayenne pepper.
Bring to a simmer over medium heat, then reduce to low and cook uncovered, stirring often, until the onion is very soft and the mixture has thickened slightly, 45β60 minutes.
Blend until very smooth with an immersion blender, 1β2 minutes, or carefully in batches in a countertop blender with the lid vented.
Press through a fine-mesh strainer back into the pot to remove skins and seeds; discard solids.
Simmer over low heat, stirring frequently to prevent scorching, until thick enough that a spoon dragged across the bottom leaves a line for 1β2 seconds, 20β35 minutes.
Remove from heat, let cool 15 minutes, then transfer to clean jars. Cool to room temperature, cover, and refrigerate until chilled and set, at least 4 hours. Keep refrigerated for up to 1 month.