In a large mixing bowl, combine the all-purpose flour, bread flour, salt, and sugar. Mix well.
Add the active sourdough starter and warm water to the dry ingredients. Stir until a shaggy dough forms.
Knead the dough on a lightly floured surface for about 8-10 minutes until smooth and elastic.
Gently fold in the raspberries and chopped pistachios until evenly distributed.
Place the dough in a lightly greased bowl, cover with a damp cloth, and let it rise at room temperature for about 4-6 hours or until doubled in size.
Once risen, punch down the dough and divide it into 8 equal pieces. Shape each piece into a ball and then poke a hole in the center to form a bagel shape.
Place the shaped bagels on a parchment-lined baking sheet and let them rest for 30 minutes.
Preheat the oven to 425°F. In a large pot, bring water to a boil and add the baking soda.
Boil each bagel for about 1 minute on each side, then return them to the baking sheet.
Beat the egg and brush it over the bagels for a shiny finish.
Bake the bagels for 20-25 minutes or until golden brown. Let cool on a wire rack before serving.