So I broke out the rest of my chemistry set and did an analysis.

Brita's special filtration resin removes calcium and magnesium ions and replaces them with H+ hydrogen ions, resulting in the acidic pH of filtered water. It does not remove carbonates, which are now present in excess as HCO3-. If I were to boil this, my pH would go from 5.8 to a likely ~10 because the CO2 gas would split, leaving OH- ions in place which would cause the rise in pH.