I have not worked with pine tar
@Mike E, but looking at your soap and the swirls in it, I would guess that your ingredients were not thoroughly mixed. Do you fully blend all your oils prior to adding the lye solution? If so, do you think you may have been unable to fully blend the lye and oils before the mix started to seize? I am very curious to see what result you get with a zap test. I would test the wet spots separately from the solid gray areas and see if you get the same result.
BTW, you made me do the math. I don’t know what size bars you make, but I guesstimated 4 ounces. That means you made at minimum 20 pounds of soap. What container did you mix it in? Are you using a drill or something else to blend your batches? What kind of mold are you using? Show us more pictures, please! We like to look.
Edit to add: What are the white spots caused by???