I intended to run depth-first beeline research through Physics - Electricity - Radio - Mass Media, but changed tack after Electricity. I went to Biology for more food and the National Park, then to Refrigeration, a surprisingly enticing tech. Cereal Mills would help a ton in my food-poor cities (we do have 6 wheats, good for +5 food), and Supermarkets were also worth +2 health for my health-constrained cities.
I traded my way through the other tech branches: Sci Meth for Constitution, Physics for Democracy, Democracy in turn for Gunpowder and Chemistry, Chemistry in turn for Corporation, Biology for Rifling and Mil Tradition and later Steam Power, which I flipped in turn for Military Science. I'd long since blown past WFYABTA, so Pericles and Victoria cut me off, but Lincoln and Justinian were both Friendly allies on Team Buddhist and kept flipping me all the techs I needed.
I did the Statue of Liberty, knocking out part of it with a Great Engineer to make sure I won it, and did adopt both Democracy civics.
On the culture front, I had four religions going for the temples and cathedrals. Since ancient times I'd had Buddhism, Confucianism (self-founded), Judaism, and Christianity, and now somebody sent me a Hindu missionary so I spread that around too. I'd never get any entry to Taoism or Islam, but didn't really need them.
After having left Hereditary Rule for Universal Suffrage (the culture cities all had lots of towns and greatly welcomed the hammers for cathedral construction), my cities ran into some happiness constraints yet again. Fortunately, there was a new game element to the rescue - Forest Preserves. My cities have lots of forest tundra tiles that we lack the food to work, so I made great use of Preserving them for the happy faces.
I picked Yekaterinburg, an otherwise unremarkable city, as my Wall Street and corporate HQ city. It had the highest commerce after the three cultural cities, which would be busy with religious buildings for forever, so was a good choice to build Wall Street and train executives. With Cereal Mills spread to my culture cities, I traded for everybody's spare wheat, to get the corporation as high as +10 food.
And now I needed to change civics again. With the National Park done, I wanted to get into endgame Great Artist production configuration: Caste System and Pacifism. Also, I had to leave Mercantilism in order to found and spread corporations. So, since this would be two turns of anarchy, I popped the 1-man Golden Age now.
Also with that, I had to run 10% culture slider again to make up for Emancipation anger, which would continue for the rest of the game. Not that running the culture slider is at all bad in a cultural victory, heh.
Pericles had made peace with Justinian a while back, but now I got another break as Genghis declared on Greece to at least keep him busy.
And that silly American tundra village in the middle of my nation finally saw the light of Russian culture.
In time, Mass Media arrived, and of course I founded Civilized Jewelers. This corp would be the star of the late game show. And I went around buying up the whole world's gem supply, paying combinations of Deer, Hit Musicals, Hit Singles, and gold per turn whenever necessary. (I'm not sure why the AI thinks there's any value at all in trading for excess Deer, but hey, everything is worth what its purchaser will pay for it.) This totaled to 17 (!) gems resources, for 68 culture per city!
(Incidentally, how exactly does this work in reality terms? This is like Paris having a jewelry company that contributes more to France's national identity and culture than the Louvre, Notre Dame, Eiffel Tower, and its operas and theaters and cinema all put together. 
)
Finally, since I had all the prerequisites, my last two research targets were Assembly Line and Industrialism. With Marines, I'd have a strong response to any potential aggression and could safely go 100% culture.
In 1854 AD, that breakpoint was reached. Each of the culture cities was already over 25,000 culture, with the math roughly being 30 from specialists + 30 from buildings + 72 from Civilized Jewelers + 400% = 650 or more per turn already. The culture slider added another 80-100 commerce to each city, for 500 more after the multipliers. And I fired a last Golden Age with two remaining Great People (an unhelpful Spy and an Engineer that I'd been saving for Creative Constructions which wasn't going to arrive).
Moscow and Yaroslavl matched each other nearly perfectly, but Novgorod overshot them both by 15,000 culture. The National Park specialists had produced zounds of culture (although that huge discrepancy is still only 8 turns worth.) It helpfully made one more Great Artist just before the end, letting me pop culture bombs in both Moscow and Yaroslavl together on the final turn.
Cultural Victory in 1880 AD.
Here's the replay, though there really isn't much to see. Almost no conquering happened anywhere, ever. No more than a handful of border cities ever changed hands.
The final GNP graph, which is extremely deceptive. It looks like a blowout, but that's fake. After that 1745 AD marker, I spread Civilized Jewelers for piles of culture, and later put the culture slider to 100%. That graph line is reflecting all my commerce plus 70 per city from the corporation piped through +400% multipliers. The real economic competition was as it was up through the 1745 AD marker, with me competitively on par but never opening an actual lead.
(I'm starting to see what Sulla means with his complaints that Alexman liked making changes to the game code and elements without thinking them through. The original Civ 4 GNP graph was misleading, because it only counted commerce and omitted specialists and all multiplier buildings and shrines. The new GNP graph includes all that, but includes too much in culture and espionage as well, which have no economic value. So we still have no true comparison of economic power between civilizations.)
39847 game points in the Hall of Fame. Wow, I expected significantly more on Immortal difficulty, not half the score of a Prince win at an almost identical date. Well, as always, that score is skewed towards skirting the domination limit to accumulate ingame score before the normalization multiplier, which the Dutch game did nearly perfectly.
Well, this game does expose how easy the cultural win has become. The BTS expansion drastically favors cultural wins in a TON of ways:
In fact, I'd call culture so easy now that it's likely to be the default peaceful win condition in BTS rather than the space race. This is even true for the AIs as well, which can sometimes beat a player's space attempt with a cultural win. I'm having a hard time seeing any situation (besides variant rule) to go for the longer slower space race rather than the easy cultural exit...?