When added to a pool that has some calcium in it, the soda ash (aka washing soda) will raise both pH and TA. It's very high locally so the saturation index is high and precipitation or cloudiness from calcium carbonate is the result.
Na2CO3 + H2O ---> 2Na+ + HCO3- + OH-
Sodium Carbonate + Water ---> Sodium Ion + Bicarbonate Ion + Hydroxyl Ion
...................................................................... +TA ................ +pH +TA
8 pounds of soda ash added to 25,000 gallons with starting pH of 6.5 and TA of 40 ppm results in a pH of 7.25 and TA of 76 ppm, but locally until it fully dilutes in the water the saturation index is much higher. In 100 gallons the saturation index (with CH 300 ppm) is +2.56, in 1000 gallons it is +2.46, in 10,000 gallons it is +1.46 so calcium carbonate is formed and doesn't dissolve until the dilution is into at least half of the pool water.
The reason this doesn't happen as much with baking soda is that it raises the TA but doesn't raise the pH as much.
NaHCO3 ---> Na+ + HCO3-
Sodium Carbonate + Water ---> Sodium Ion + Bicarbonate Ion
...................................................................... +TA
12.7 pounds of baking soda (for the same TA increase as the above soda ash) added to 25,000 gallons with starting pH of 6.5 and TA of 40 ppm results in a pH of 6.77 (ignoring carbon dioxide outgassing) and TA of 76 ppm, but locally until it fully dilutes in the water the saturation index will be somewhat higher, but not nearly as high as with the soda ash. In 100 gallons the saturation index (with CH 300 ppm) is +1.79, in 1000 gallons it is +1.07, in 10,000 gallons it is -0.37 so there is far less calcium carbonate formed.