Wow, glad I asked! I thought this was an easy decision, but it looks like we are split.
If I'm interpreting correctly... The-Hawk, markh, and Pariah are in the "keep it" camp. drhirsch and AgedOne are in the "raze it" camp. Sam started in "keep it", not sure he is still there.
The "raze it" gang raises some good points. If we keep Corinth, we risk severly stunting our already stunted tech rate. In fact, it is possible that Corinth bankrupts us at our current military level. Corinth and Athens together might be even worse (although Athens seems to have some cottages and may break even).
However, Corinth is a big prize. If we capture Stonehenge, then we will get a shrine for free. Given Corinth has two food bonuses, we should be able to whip a monestary and maybe even a missionary or two before Cyrus pillages our fish and crabs. Once we get IW, the iron mine will make Corinth a decent production city. Remember, we are going for dom... at some point, we will have a bunch of cities on lush island. I think the shrine pays for itself in the long run. Given war weariness will be a major factor, the happy face from a religion is a big plus. And, don't underestimate the value of Theocracy for our later military.
Here is a third option we might consider: Maybe we should leave Corinth alone for a while. We pillage the Athens to give Alex something to do (but not the cottages! we want them!). Then we make peace (hopefully for a tech if he has Alpha). We tech to CoL, then take both Athens and Corinth. In other words, we delay the economic hit until we have the ability to build courthouses.
My issues with the third option: We are low on gold and teching at 20%, so it is going to take forever to get to CoL. Meanwhile, Alex will resettle the silver island. Not a huge problem since we have military there, but it means we are almost back to square one. This option is less risk, but sets us back a ton of turns.
If I were playing this myself, I would grab Corinth and Athens asap, then worry about getting my economy back to life. It will be a struggle, maybe a disaster, but I think a cautious approach simply means we lose to the other SGOTM teams. Of course, as an HOF player I am conditioned to take big risks, maybe this approach is too risky for SGOTM where we can't spin up another game if this one flushes down the tubes.
Side note: I was planning to follow Constr with IW. However, this dilemma shows we need to get CoL next. The fact that we are considering razing a city like Corinth shows we need to get courthouses before we can do any serious conquest.
Captain Mark.... I think this decision warrants more conversation, so I'd like to delay another 24 hours before I finish my turnset. This will allow another round of points and counter-points before we make a final decision.