Okay, I wasn't done playing Sacred Sites games after the Brazil game in 109 turns. Redditor DushkuHS suggested that Spain with a sufficiently perfect start may be able to beat even Brazil's Carnival to the culture victory. Naturally, I had to try that.
With a faith-producing natural wonder, Spain can produce the pantheon and religion on a whole new level of speed than any other possible method. There are three natural wonders that accomplish this: Mt Sinai (8 faith), Uluru (2 food 6 faith), and Sri Pada (2 food 4 faith 2 happy). Of them, Uluru seems best, although any will work. The Great Barrier Reef is also a possibility, with no faith production but massive yields in other areas; it wouldn't accelerate the pantheon but then via One With Nature could accelerate the religion.
So launching this game involved start-scumming as usual, but to be honest, also just a little bit of cheating. My process was this. I rolled maps repeatedly, walking the starting settler and warrior around for ten turns or so, looking for one of the aforementioned natural wonders. Once I found one, I would reload to 4000 BC and walk my starting settler there directly. I tried to ignore any other map knowledge other than the location of the natural wonder.
The whole process took about an hour. About half of the maps would find no natural wonders at all, and most of the rest would find one of the crappy ones like Barringer Crater. One map did have Sri Pada within range, but in a terrible location out on a peninsula with no other food or good tiles. But finally I got this.
The instant I found Uluru, I reloaded from the initial autosave, trying to keep in mind nothing else about the map layout. On the reload, I walked the capital settler directly to Uluru, and otherwise scouted just by instinct.
Yes, this is cheating by CFC and Realms Beyond rules. I know that. But it's only a shortcut. I could get the same results by rerolling maps until the initial settler happened to walk directly into Uluru. It's just saving my sanity on the amount of map rerolling. This is an experiment, not any kind of serious competition.
Anyway, this is a fantastic start. Uluru is in a fine location, with some resources, and also enough desert to use Desert Folklore over One With Nature. Uluru produces 12 faith per turn when doubled by Spain's ability, so that got the pantheon right away in a single turn on turn 4. Desert Folklore would be less faith than OWN for a while, but would ramp up fairly quickly.
Also in the picture is another natural wonder, Krakatoa. I actually didn't even find this before Uluru, it came by honest scouting after the reload. It wouldn't be workable for quite some time, but it does yield a second shot of 500 gold.
So, where do we spend the gold? First we had to buy the Uluru tile itself, which is why there's 915 gold and not 1000 in the treasury. Next, notice that there is no tile at Madrid that a worker can help yet, we need Animal Husbandry or Calendar, so I put off the worker for a bit. Scouts looked to be the most time-sensitive item, with the most to gain by getting out a few turns sooner. So I bought three of them, and a settler (on turn 7 after Madrid grew, you can't buy settlers at size 1. The ideal Spain start involves a ruin for population to size 2 immediately, but I didn't get that here.) Like in the Brazil game with El Dorado, I bought the other stuff and set the monument as the build order, since that's the worst gold:hammer conversion for purchasing.
My first research here is Mining, because my bought settler would plant directly on the silver to sell it right away, which happened on turn 13.
There are two ruins visible at Madrid in that shot. The western one popped culture on turn 5, great. Beyond that, I did not get a lot of useful ruins, even despite the three bought scouts. I got barbarian camps from ruins like four times. The useful ruins were gold, gold, Archery, and culture again a bit late (t34).
A neat note: Uluru is impassable as a mountain, so it's safe and can't be blocked by wandering barbarians standing on it.
Here's an overview 20 turns in, as we've explored our neighborhood. This is enough good stuff to work with, a good amount of desert. And just enough luxuries, four distinct ones (Barcelona is founded on silver) once we get down to the crabs, and extra sugars to trade for more.
Build order at Madrid, after buying the three scouts and settler: monument - scout - worker - settler - settler - archer - shrine - settler - settler. I bought the worker at Barcelona with the usual gold from city-states and also a later natural wonder find.
Research order: Mining (silver) - Pottery (granary in Barcelona) - Calendar (three sugars) - Animal Husbandry (two sheep at Madrid) - Bronze (find iron) - Writing - Philosophy.
That picture has my Great Prophet, on turn 20 by far the fastest I've ever seen and just about the earliest possible. He founded the religion, for my now standard picks of Initiation Rites and Pagodas. The enhancement prophet followed on turn 38, also about as fast as possible, taking Mosques and Religious Texts. I thought about taking one pagoda before enhancing, to snowball faith a bit faster, but didn't want to risk missing Mosques to someone else's religion who had Stone Circles already. Also Religious Texts sooner would convert cities sooner and make up via Desert Folklore most of the faith that the pagoda would have produced. On t36, Barcelona grew to size 4, where Desert Folklore was producing the same 8 faith on par with what One with Nature would be doing. DF had foregone about 120 faith compared to OWN early on, but would indeed make that back up pretty quickly from there.
I continued to sell resources as they were improved (sugars on turn 30 and 33), and also Egypt and Venice friended me to borrow gold. I bought my next workers on turns 27, 37, and 47, then more later as new cities expanded.
Social policies: One significant advantage of Spain over Brazil is not needing to put policies towards Representation. As per suggestions from DushkuHS in that Reddit thread, I decided to go down the left side of Liberty, where the great ICS policies are. I got the Liberty opener on turn 8 via a ruin, t16 Republic, t27 Collective Rule. That's seriously fast. I've complained before that Collective Rule comes just a bit too late after it was nerfed to go behind Republic, typically turn 40 just after you've wanted to build a couple settlers. But it came fast here thanks to two early monuments. Spain does accelerate in many ways across the board.
I got some good CS quests -- three for finding Maya lands! -- two of them militaristic that gave very helpful units (archers).
And a look 50 turns in, at the astronomical acceleration achievable by Spain. Compare this shot to Brazil on turn 68. These shots are almost identical; Spain has accelerated by almost exactly 18 turns. Same six cities, close in total population at 17 vs 20. (Madrid didn't grow like Rio because it was building settlers via Collective Rule instead of buying them.) Freaky identical 27 vs 26 faith production, 354 vs 366 scoreboard points, 17 vs 16 culture per turn, and 142/230 vs 140/230 culture! But Spain is also ahead by one pagoda already and halfway to a second.
I followed my now standard plan of holding fledgling cities at size 1 until they convert. Seville was first and it quickly snowballed to all the rest from there.
We can also compare the civs on equal terms at the same date of turn 68, when Brazil enhanced the religion. Spain is way ahead. 9 cities to 6. 34 population to 20. 70 faith to 26. 526 scoreboard to 366. 39 culture vs 16. 6 pagodas to Brazil's 0.
How did we get that far ahead? It's not just Spain's natural wonder gold and yield, that accelerates well but not that much. I think the key difference is the ICS policies in Liberty, Republic and Collective Rule. I said before that Sacred Sites needs ICS but not fast, but maybe it does. Expansion begets faith, by both Desert Folklore and also all the shrines and temples add up considerably. Republic is seriously important in building those ~10 turns faster. And expansion even plays into Initiation Rites that steadily fed money to buy workers for the new cities.
So it's a meaningful weakness for Brazil to have to spend policies towards Representation instead of towards Collective Rule. "Why not both?" There isn't time, you can't put five policies into Liberty and still get Sacred Sites by turn 100.
But Liberty was now past, and I was developing Piety. I debated whether there was enough time to squeeze in Mandate of Heaven, but eventually decided that there was and I think that was correct. t40 Piety opener, t56 Mandate, t69 Organized Religion, t73 Religious Tolerance via Oracle, t79 Reformation for Sacred Sites.
So at 26 tourism per turn on reaching Sacred Sites, what does the victory date say? 42 turns, that's turn 122. Spain has no Carnival to work miracles here, but they do have faith snowballing far ahead of what Brazil was managing. Spain still has headroom to double that tourism with more buildings, and I could see that would get the victory date down pretty close to Brazil's.
I got one more policy beyond Reformation, on turn 89. The most useful choice looked to be Theocracy for the Piety finisher for the Prophet. Faith from the holy site would speed up more mosques by a turn or two and perhaps shave the end date.
The one thing that went wrong in this game happened now, when I lost the Parthenon by four turns. Dammit. I just never get that wonder, even though I could have sworn I started it early enough here (turn 73 right after the Oracle.)
One more overview to compare with Brazil, on turn 90. Spain is ahead 14 cities to 7. 56 population to 33. 843 scoreboard to 555. 121 culture vs 34 holy crap how did that happen. 42 tourism vs 18 - yes Spain is by brute force more than doubling up Brazil so even overpowering Carnival.
I also have a city working Krakatoa down there, 10 beakers with Spain's doubling. I was doing this just for the heck of it, but it actually became significant. I reached Civil Service several turns faster thanks to that, and got Open Borders with the leading culture civs to speed the endgame.
And there we are, got the victory down to exactly 100 turns, 375 BC. (Actually a bit faster considering Madrid didn't start until turn 3.)
So yeah, perhaps indeed Spain's acceleration can beat Brazil's Carnival. You go from Sacred Sites to victory in just 20 turns anyway, so Carnival can only cut off 10. Spain can speed up by more than that on the front end.
It's the same principle as for science games, accelerate the front end not the back end. The constraint for science is to finish your universities and work scientists as soon as possible, not to go from universities to endgame which turns out to be essentially incompressible. Same goes for culture: the constraint is getting the mosques and Sacred Sites in play quickly, not going from them to endgame.
I actually might have one more approach for a culture game in mind, stay tuned. 