John-
I re-read your original post; and it still looks to me like the tug operator was just trying to do his best to avoid you by turning to starboard. Maybe he was also trying to stay outside or on the edge of the shipping lane for safety and he figured that if he turns to starboard (as required) and you turn to starboard or maintain course you would pass port-to-port without problem. Tugs often don't (nor are they required to) stay in the shipping lanes; so long as they are not at risk of going aground. It's possible he thought you were another tug operator or commercial boat; and the routine traffic alongside the "lane" is port-to-port when it's foggy.
The hard turn to Port after he crossed your stern is clear indication that the tug operator saw you on his
radar and was trying to avoid running into you (he was returning to his previous course after getting clear of you). I think your hard turn to starboard when at close range was the correct thing to do to avoid the collision; next time try easing over to starboard before it gets to a close quarters situation. I would not expect other large boats to be in close proximity to the starboard of the tug (unless the tug had a tow or ship alongside); in which case you should see it on the
radar.
I think if you look at your track data you might find that your turn(s) to port put you further from the shipping lane; and if you had stayed on your original course you would have passed the tug port-to-port without a major course correction. He likely would have eased to starboard and went by with plenty of room.
I don't know what the exact speed of the tug was but it seems that he should have slowed down more (but hard to say he might have slowed down but you did not see it). I still think his turns to starboard were to avoid collision (as the regs require) rather than to cause one. Tug operators tend to be VERY aware of the navigation regulations and follow them to the letter. It's a long tough road to become a tug captain; and turning to port when approaching another vessel in the fog would be an unwise career move for them in -most- situations...