Prep fruit & zest — Zest one blood orange and juice 2–3 blood oranges until you have about ¾ to 1 cup juice. Rinse cranberries and discard any stems or bad berries.
Bloom sugar (optional but helpful) — In a medium heavy saucepan, stir together the sugar and 2 tbsp of the orange juice (or water) until moist.
Add main ingredients — Add cranberries, remaining blood orange juice, the cinnamon stick (or ground cinnamon), sliced ginger (if using), and a pinch of salt. Stir to combine.
Bring to simmer — Place over medium-high heat and bring to a rolling simmer. Cranberries will begin to pop, releasing pectin.
Simmer & adjust texture — Reduce heat to medium or medium-low. Simmer gently for 8–18 minutes, depending on desired texture. For chunkier sauce, aim 8–10 minutes; for jammy, mashed consistency, 12–18 minutes, mashing gently with a spoon or masher.
Remove aromatics & thicken if needed — Discard the cinnamon stick and ginger slices. If sauce is still too loose, stir in the cornstarch slurry and heat 1–2 minutes until glossy and slightly thickened.
Finish flavors — Remove from heat and stir in the orange zest and optional liqueur (Grand Marnier, Cointreau, or ruby port). Taste and adjust with a pinch more salt, extra sugar, or a dash of vinegar if brightness is needed.
Cool & set — Transfer to a bowl or jar, cover and refrigerate. The sauce will thicken further as it cools. Stir or reheat gently before serving if needed.