Write a descriptive and concise alternate history scenario in the form of a lengthy history book chapter in which Fukumatsu, the legendary Ming loyalist general who would become famous for his prowess in battle against the Manchu armies during the Manchu conquest of China in the 17th century under the name Zhu Chenggong/"Zheng Chenggong"/īⁿ Sêng-kong aka Koxinga, never goes to his father Zheng Zhilong's homeland of Ming China when he is seven due to the breakdown in communications between the Ming dynasty and the Tokugawa shogunate, and is instead raised as one of the princes of the House of Yamato, the imperial family of Japan thanks to Fukumatsu's mother, Tagawa Mastsu, being a descendant of the House of Yamato, Japan’s imperial family through Japan’s 50th Emperor, Kammu Tenno, through Taira no Shigemori and Tagawa Yazayemon. 
Fukumatsu still ends up on the side of the Chinese imperial forces fighting against the Qing Empire's conquest and consolidation of China due to him getting mixed up with Ming remnants during a visit to his father's hometown in Fujian, but thanks to his military experience and training as a samurai in his homeland of Japan is able to lead the Ming remnants in Southern China to more conclusive victories, giving the Nan Ming [Southern Ming] enough time to escape to Indochina where it [the Nan Ming] proceeds to overthrow the native Le dynasty of Vietnam and become the new government of Vietnam, conquering the Khmer and Laotian Kingdoms and making them into satellite states. 
Fukumatsu's foundation of the Dōngníng Wángcháo [Taiping Kingdom] on Taiwan in 1661 after liberating the island from Dutch colonial rule is to help as much Ming loyalists and anti Qing Han Chinese escape to Indochina as possible, and when this is complete he is able to use his status as a member of the Japanese royal family to have the Dōngníng Wángcháo integrated into Japan as a region of the Tokugawa shogunate, in the hopes of reclaiming the Korean peninsula which was lost by Japan when rouge military commander Yi-Seong-Gye proclaimed the Joeson Kingdom in 1394, ending thousands of years of Japanese control over Korea, which ironically does happen thanks to Fukumatsu's successful campaigns against the Qing Empire leading to the collapse of the Ming's trusted ally in the Joeson kingdom, allowing the Japanese to reluctantly regain control of the Korean peninsula from the weakened Yi dynasty.
By the time of Fukumatsu's death in 1672, the Nan Ming has consolidated its control over Indochina and has even regained parts of Southern China, while the Qing Empire is able to repopulate the regions of China left empty by the migration of Han Chinese to Ming controlled Vietnam with national minorities and focus on improving and developing its conquered territories, and the Japanese shogunate has become the de-facto successor of the Turkic Tang Empire which ruled China from 618 to 907, with many nations in South East Asia submitting to the Japanese for fear of Manchu expansion and Ming revanchism.
Dōngníng [Taiwan] and Chosun [Korea] remain Japanese up until the 21th century and beyond, thanks to the Tokugawa shoguns exploitng the chaos in the aftermath of the Ming's demise to remove yangaban and Yi dynasty influence from Korea as well as liberating the nobi [slaves] and gisaeng [sex slaves of yangaban] and promoting social reforms that also reduce the Korean identity to being part of the greater Japanese nation.