Do yourself a favor and look for Fuyu persimmons and tangerines that are seedless – it will make the fruit preparation go a lot faster. I like Satsuma mandarins as they are reliably sweet and easy to peel, but other varieties will also work, as probably will navel or Cara Cara oranges. Peel the tangerines and separate them into segments. Cut the persimmons into large chunks or wedges. Freeze both on a rimmed baking sheet, then place them in bags or containers for storage.
Combine the cashews and almonds in a 1-quart jar and fill the jar with water. Let soak 8-24 hours. Drain the nuts and rinse them well, then add them to the blender with 4 cups of water. Blend on high until very smooth, about 3 minutes. Strain the mixture through a nut milk bag and into a large bowl, squeezing the dickens out of the pulp to extract as much liquid as possible. (Reserve the nut pulp for making crackers or brownies.) Add the remaining 6 cups of water to the nut milk (or add less water for a richer milk). The nut milk will keep, refrigerated, for up to 4 or 5 days.