In a large bowl, mix the sourdough starter, warm water, and bread flour until a shaggy dough forms. Let rest for 30 minutes (autolyse).
Add salt and knead until the dough becomes smooth and elastic.
Divide the dough into two equal parts. Knead beetroot powder and crushed raspberries into one half, and pistachio paste into the other half until well incorporated.
Place each colored dough in a separate bowl, cover, and bulk ferment at room temperature for 4–6 hours, or until doubled in size.
Roll out ropes from each dough, twist one raspberry and one pistachio rope together, then shape into a ring to form each bagel. Place on parchment-lined trays.
Cover the shaped bagels and cold-proof them in the fridge overnight (8–12 hours).
Preheat oven to 425°F (220°C). Bring a large pot of water to a boil and add honey. Boil each bagel for 1 minute per side.
Remove with a slotted spoon and place back on the tray. Sprinkle with chopped pistachios and extra raspberries if desired.
Bake for 20–25 minutes or until golden and slightly firm on top. Let cool completely before serving.