Since the 15 percent is listed as water, this does not explain the white precipitate unless there is something else in the A&H material that the MSDS does not mention. For what it's worth, I looked at several specifications for industrial Na2CO3 and they generally state that the product contains less than 1 percent insoluble matter.
Sodium carbonate, like many other salts, can contain chemically bound water molecules. When I looked at the A&H Washing Soda MSDS and saw the water content, my assumption was that the product is probably a blend of anhydrous Na2CO3 and Na2CO3 with bound water. This is consistent with General Chemical's technical data, which states that at temperatures below 109C, the solid phase can exist at about 85 to 100 percent Na2CO3, and the balance being Na2CO3 with one water molecule. When Na2CO3 (with or without bound water) is dissolved in water for electrolysis, the concentration of the solution is below - a lot below - the saturation point, so there would be no white precipitate. The soda ash all goes into solution, yielding sodium ions and carbonate ions.