From your pic I can’t tell what is protruding. To my eye it looks like mucus but that may just be the effect of zooming the photo.
I am NOT a fish medic but I do have fowl and parrots and the reproductive and elimination anatomy is similar. Eggs/fry, poop and urea all exit the same place. It is not uncommon for avian species to become egg bound (not even suggesting this is what’s happening with your tang). When this happens in fowl, we try HARD to do everything we can to facilitate passing the egg naturally. When lethargy and anorexia set it most of us go the extreme route with manual manipulation and even breaking the egg and extracting shell. I am 60/40 on the positive side with manual intervention of saving the animal. I am 0% success rate with just watching and waiting and doing less invasive methods when the animal hits lethargy an anorexia. I also deal with retained placenta in my large livestock and if there is something hanging out, I’m pulling to at least test resistance. If you meet resistance, stop.
Because of that experience, if it were MY fish, and it is not, I would at least see if I could gently manually remove anything I could see. The likelihood is it’s just going to disintegrate as soon as you pull anyway.