Thanks for the tips. This was more of an "opportunistic" diplo win chance. I didn't have much of a strategy... earlier in the game Hammumbari had attacked me, so I converted to Buddhism to please Qin, so I could bribe him into war. Then got everyone else to pile in. I had the opportunity to vassalize Hammumbari but thought I'd be better off taking over his cities... bad mistake since Qin vassalized him instead a couple of turns later.
After that it was a love-in with no opportunity for military conquest or setting civs off against each other, since we were all friendly and signed defensive pacts. And I was behind in both military and tech, but had quite a lot of Ham's old cities (actually mainly those of his now-smited vassal, Catherine), which put me in the top 2 population-wise. So the diplo opportunity was a spot of luck. I did have Qin on +3 trading fairness and forthrighteousness, mainly from simply gifting him 1000s of gold since he was ahead on tech. Switched to Bureaucracy (from Free Speech) to please him, and also to Hereditary Rule to please Ragnar. Ragnar was also Hinduism/Theocracy, so I bribed him into Organized Religion, sent some missionaries, and got him to change to Buddhism, which made him like me enough to vote for me on the final time (above). Didn't think of liberating a city, but will know for next time.
Anyway, diplo didn't work out, so I passed the Free Religion resolution in the hope that people would like each other a bit less and cancel their defensive pacts. No such luck. It was a very corporation-y end game. I had Creative Constructions, which was pretty useful, but it looked certain that Huayna was going to win Cultural.
However by now my extra land area had given me the biggest industrial and tech base, and by ignoring military I was able to get the space ship built, but it looked like it would be too late since Huayna had been building cultural stuff and already had 2 Legendary cities, plus Ollantaytambo on 70k (out of 75k). So the final 30 or so turns were spent on 100% Spy, producing 2000 points per turn, all directed at Huayna, with about 10 spies in his city, supporting revolts every turn. First time I've used this strategy, as I haven't normally used spies for much other than scouting about, and nicking technologies after sprouting a Great Spy early on. It worked a treat, so I was pleased with myself for thinking this up (desperate times...). Was up to -12 with Huayna for spies caught causing trouble, and he eventually cancelled all his deals with me, but there was still nothing he could do about it.
With 3 turns to victory, Qin launched a last ditch comedy invasion with a huge stack of modern armor/mobile artillery/stealth bombers etc and pwned one of my cities, but of course he'd left it far too late and I claimed a hard-fought space-race victory. (Monarch difficulty / K-mod / Epic Speed / PerfectWorld_2 map)