The key is that Potassium bitartrate (KHT) is part of a buffer system and is an ion that can be an acid or a base depending upon the pH. If you think pH is hard to understand, buffers are even uglier, but I'll try to do it here without getting to technical.
At low pH buffer, KHT is a base, and tartaric acid (HHT) is the acid, so removing a base by precipitation lowers the pH of a solution. At high pH buffer, KHT is an acid, and dipotassium tartrate (KKT) is the base, so removing an acid by precipitation causes the pH to increase. The tipping point between the high and low pH buffers is around pH 3.9, but temperature and amounts of acid and other ions can shift this slightly ~ 0.1
Check out p7 of the link in post #4 for a more visual explanation of the buffer components, and why pH can change in both directions.