This is a very hasty sketch I threw together about an interesting idea I've been mulling over for a while. A timeline in which Southern Ming survives...
In this timeline, Ma Shiying decides to lead an army against Zuo Liangyu himself, but is defeated and killed. Zuo Liangyu then enters Nanjing and whips the Ming court into shape, firing corrupt officials and rallying the princes around the Hongguang Emperor. The death of Ma Shiying means that Shi Kefa isn't diverted to attempt to defeat Zuo as IOTL, and is able to consolidate the defences of Yangzhou. When Prince Dodo attempts to attack the fortress, his force is annihilated and the Ming are able to hold the south bank of the Yangtze against Qing attack. Further attacks on the river prove fruitless and eventually the two states settle into an uneasy peace.
After the bitter disappointment in the south, the Qing focus on the north, and when, in 1670, the Southern Ming Jianshun ( 建順 ) Emperor decides to focus on shipbuilding to protect the Yangtze and block Qing sea trade, the Qing decide to embark on a shipbuilding program of their own. In preparation for this, they conquer and colonise areas of southern Siberia to obtain a steady supply of shipbuilding hardwood, and, under the Guangwu ( 廣武 ) (OTL Kangxi) Emperor successfully conquer Korea in 1683 to obtain nautical experts and infrastructure.
However, the resources expended on South China and Korea have weakened Qing's grip on the west. The Dzungar Khanate doesn't decline and remains a formidable force. Butterflies prevent the civil wars that plagued it later on. In 1715, the Dzungars would conquer Thibet, taking advantage of a succession crisis among the Kalmyk rulers there, and would move the capital to Ningxia. Later on, the Dzungar Khan Zobda (an ATL grandson of Tseten Khan), who became enamoured with Chinese culture after the Southern Ming sent emissaries and gifts as overtures for an alliance against the Qing, would declare himself the Tianquan ( 天權 ) Emperor of the [later] Shang Dynasty.
In 1740, the Qing Empire invades the Shang. After defeating the initial assault, the allied Ming and Shang Empires launch a coordinated campaign against the Qing, and in the Battle of the Three Emperors, so named because the Hengjing ( 亨景 ) Emperor of Qing, the Yongdao [ 雍道 ] Emperor of Southern Ming, and the Hongwen [ 弘文 ] Emperor of Shang directly commanded the armies on the battlefield, the Ming decisively defeated Qing. However, the Yongdao Emperor knew that Southern Ming, whose military was geared towards defence, would become overstretched and could not fight the Qing for long in an offensive war, and immediately called for peace negotiations. The Southern Ming and Shang both gained territory.
The Yongdao Emperor would rule for 21 more years of peace, and would work to strengthen the Southern Ming economy and culture. His heirs, however, would not be so wise. In 1763, his grandson, the young Zhuode ( 卓德 ) Emperor, influenced by the minister Jie Suifeng, would resume the war with the Qing. This war would grind on as a bloody stalemate, and would end inconclusively. The two empires would fight one another off and on for the next 70 years, still with no conclusive results. However, in the 19th century there would be a new development.
European powers would supply weapons to either side of the conflict, reaping great profits.
In 1830, the Southern Ming decisively defeat the Qing Navy in battle, blockading Qing ports and securing the oceanic trade. Using these profits, they buy large quantities of cannon and rifles, as well as hiring British and French military trainers. Using this advantage, the Southern Ming Xiangyi ( 祥怡 ) Emperor launches a new campaign northwards. Russia invades Manchuria from the north. The Qing break under the pressure, and by 1837 the Ming have taken Beijing. The Xiangyi Emperor changes his era name to Xuanjie ( 宣捷 ) when Beijing is taken. Russia conquers and annexes all of Manchuria.
This victory would be short-lived, however. In 1838, the triumphant Xuanjie Emperor betrays his ally and invades Shang to reclaim Qinghai and Thibet. This war degenerates into a bloody stalemate. The Ming eventually begin advancing again, but progress is still hard and resisted.
Further misfortune occurs when Ming authorities attempt to block British shipment of Opium into China. This precipitates the Anglo-Chinese War, in which the UK defeats Ming and occupies Canton and Hong Kong ports. To save the war effort against the Shang, the Ming are forced to surrender and give up Hong Kong, Hainan, and Formosa, and open all ports to European trade.
The imperial coffers are now spent after fighting three devastating wars, and the people have grown weary. When the neglected levees break and the Yangtze river floods in 1840, the peasants in the area revolt. The "Black Turban Rebellion" quickly spreads, and mutinous army units join the rebels. The Ming forces rapidly disintegrate, and the rebel leader, Yan Yijiao (a peasant turned small magistrate who passed the civil service exams), declares himself the Dachang ( 大昌 ) Emperor of the Great Shao ( 韶 ) Dynasty. The army negotiates peace with the Shang in the west, who, also exhausted from the war, agree to cede territory and resume the old alliance.
The Shao Dynasty has an uncertain future. The circumstances of its formation have allowed Russia to expand massively, and that empire is eyeing Shao's Shang allies jealously. The British and French have recognized Shao on the condition that they keep the current trade agreements, but the Europeans will not be satisfied with this for long. The demographic and infrastructural destruction wrought in China in last 200 years is gargantuan, and the Shao must work tirelessly if they wish to keep Europe and Russia at bay long enough to rebuild.