It's difficult.
Qing's advantages
The Qing control of China was nowhere as loose as the Yuan. At least the empire did not fracture into several warlord kingdoms immediately after the the initial uprising.
During the End of Yuan rebellions, rebels under different flags set up kingdoms up and down the Yangtze River. The Mongol military machine was in the state of complete disuse, so much so that the rebels even ventured into Mongolia and set Xanadu on fire.
The same thing did not happen to the Qing. At the very least, the Confucian gentry was still loyal the Qing, so were the Mongol princes. More importantly, both of the two forces were in good enough shape that they proved capable of raising an army for the Qing Emperor.
Ideology
If the Taiping rebels wants to win, they must first adjust themselves ideologically and fight for the sympathy of the Confucian gentry. Almost all Chinese rebellions had some connection to a bizarre or unconventional religion, but all the successful ones shed off their original cultish color.
Geography
Next, they must find a good base for their Kingdom. Lower Yangtze plain has never been such a place, as it was a not an easily defensible terrain, and too close to foreign interests. The most ideal location for the Taipings was, IMO, Hunan and Hubei provinces. If they could carve out an area large enough, they could be connected to Nian Rebels and Muslim Rebels in the North, their place of origin in the South, Sichuan in the west and Lower Yantze in the east.
Hereafter
The goal was always to spread the rebellion more widely over the empire, so that the different rebel forces (Nian, Black Flags, Tiandihui, White Lotus, Muslims etc.) could work together to bring the dynasty down (before they themselves, inevitably, descend into a Civil War.)
It's doubtful that the Manchus could organize a Northern Qing (too many Manchus were caught in Chinese inland), although the Mongols might be able to. My guess is that if the Qing or the Russians took Manchuria, Inner and Outer Mongolia, and Xinjiang, the Taiping would still lay claim to them and try to take them, just to make sure that the Qing was officially and completely dead.
Inclusiveness
In conclusion, winning, by the definition of "Unifying the realm again after defeating the Qing", meant that the Taiping must be extraordinarily inclusive and assimilating. They must be able to tolerate and absorb firstly the Confucian gentry and ideology, be actively recruiting them into the rebel ranks; and secondly different rebel factions and surrendered Qing mandarins; and thirdly the non-Chinese princedoms in Xinjiang, Mongolia and Manchuria.
This pretty much required the Taipings to be the antithesis of its OTL self, which could not even sort out their own theological differences.
Aftermath
What China looks like at the end of ATL Taiping wars is to anybody's guess. With the technological conditions of that era, it's doubtful that they could implement so many reforms they claimed that they would implement. But if the important ones (land and opium) were carried out, they Taiping could easily outperform the Qing in some areas.
The death toll might be smaller than OTL, consider how much killing has been done by the Qing, rather than Taiping, troops.
Qing's advantages
The Qing control of China was nowhere as loose as the Yuan. At least the empire did not fracture into several warlord kingdoms immediately after the the initial uprising.
During the End of Yuan rebellions, rebels under different flags set up kingdoms up and down the Yangtze River. The Mongol military machine was in the state of complete disuse, so much so that the rebels even ventured into Mongolia and set Xanadu on fire.
The same thing did not happen to the Qing. At the very least, the Confucian gentry was still loyal the Qing, so were the Mongol princes. More importantly, both of the two forces were in good enough shape that they proved capable of raising an army for the Qing Emperor.
Ideology
If the Taiping rebels wants to win, they must first adjust themselves ideologically and fight for the sympathy of the Confucian gentry. Almost all Chinese rebellions had some connection to a bizarre or unconventional religion, but all the successful ones shed off their original cultish color.
Geography
Next, they must find a good base for their Kingdom. Lower Yangtze plain has never been such a place, as it was a not an easily defensible terrain, and too close to foreign interests. The most ideal location for the Taipings was, IMO, Hunan and Hubei provinces. If they could carve out an area large enough, they could be connected to Nian Rebels and Muslim Rebels in the North, their place of origin in the South, Sichuan in the west and Lower Yantze in the east.
Hereafter
The goal was always to spread the rebellion more widely over the empire, so that the different rebel forces (Nian, Black Flags, Tiandihui, White Lotus, Muslims etc.) could work together to bring the dynasty down (before they themselves, inevitably, descend into a Civil War.)
It's doubtful that the Manchus could organize a Northern Qing (too many Manchus were caught in Chinese inland), although the Mongols might be able to. My guess is that if the Qing or the Russians took Manchuria, Inner and Outer Mongolia, and Xinjiang, the Taiping would still lay claim to them and try to take them, just to make sure that the Qing was officially and completely dead.
Inclusiveness
In conclusion, winning, by the definition of "Unifying the realm again after defeating the Qing", meant that the Taiping must be extraordinarily inclusive and assimilating. They must be able to tolerate and absorb firstly the Confucian gentry and ideology, be actively recruiting them into the rebel ranks; and secondly different rebel factions and surrendered Qing mandarins; and thirdly the non-Chinese princedoms in Xinjiang, Mongolia and Manchuria.
This pretty much required the Taipings to be the antithesis of its OTL self, which could not even sort out their own theological differences.
Aftermath
What China looks like at the end of ATL Taiping wars is to anybody's guess. With the technological conditions of that era, it's doubtful that they could implement so many reforms they claimed that they would implement. But if the important ones (land and opium) were carried out, they Taiping could easily outperform the Qing in some areas.
The death toll might be smaller than OTL, consider how much killing has been done by the Qing, rather than Taiping, troops.
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