Sounds, and looks, more likely to be an inorganic phosphate. These will precipitate with alkali and redissolve with acid because acid phosphates are more soluble than basic ones. Magnesium phosphate is very easily precipitated with ammonia, for example.
A brief calculation based on the amount of phosphoric acid used should be instructive in this respect, when compared with the amount of bark used - which was how much MHRB, incidentally?
The pure phosphate is nominally white but the grey colour arises through adsorption of the dark coloration from the basic MHRB soup. I bet you could get nice white crystals by cleaning up the brown vinegar solution with activated charcoal, filtering then slowly adding concentrated ammonia (if you have magnesium phosphate) or just repeating the NaOH precipitation if procuring ammonia seems like too much trouble.
Tannins would be soluble in alkali.
The precipitate is presumably insoluble in naphtha and other [neutral] organic solvents?
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