Emperor at home, king abroad
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Emperor at home, king abroad (外王內帝) was a system of conducting relations between states within the East Asian cultural sphere. Rulers of non-Chinese regimes would use the title of emperor (皇帝, or other equivalents) domestically and adopt the title of king (王) when dealing with China.  This system was applicable to Japan, Korea and Vietnam, among others.
As China was a hegemonic power in East Asia for a large part of history, surrounding states were compelled to pay tribute to Chinese emperors in exchange for peace and political legitimacy. In this system, non-Chinese regimes accepted Chinese suzerainty and acknowledged the Chinese emperor as their nominal overlord. Since Chinese emperors claimed to be the Son of Heaven and held supremacy over all under Heaven, rulers of foreign regimes were to use titles subordinate to emperor. The same doctrine also maintained that there could only be one emperor at any given time.
During the Sui dynasty, the Japanese diplomat Ono no Imoko delivered a letter by Prince Shōtoku to the Emperor Yang of Sui which claimed the Empress Suiko as "the Son of Heaven where the sun rises", implying an equal status between the Japanese and Chinese monarchs. The Emperor Yang of Sui was angered by such a claim. Since then, the Emperor of Japan has started to use the title "tennō" (emperor) for both domestic and foreign countries, and the title "king" (國王) was sometimes used for trade with China by Shogun, who was appointed by the Emperor of Japan, and influential person in the Imperial Court. China did not officially allow Japanese Emperors to use the title "tennō" (天皇) which means emperor, but China could not stop Japanese Emperors from calling themselves emperor (天皇).
During the Tang dynasty, Japanese rulers were conferred the title King of Japan (日本國王). In 894, Japan abolished the Japanese envoys to Tang Dynasty China and ended the relationship with China, which used the title of king.
During the Yuan dynasty, the Emperor Shizu of Yuan demanded the King of Japan to submit to him. In this instance, the King of Japan referred to the Japanese emperor. Japan rejected this demand, which led to Mongol invasion of Japan.
During the Nanboku-chō period, Prince Kaneyoshi refused to accept the title of king granted by China, and killed seven Chinese ambassadors in retaliation. Ashikaga Yoshimitsu, a shogun in the 14th century, was given the title of "King of Japan" (日本國王) by Yongle Emperor because he needed to trade with the Ming dynasty.
During the rule of the Tokugawa shogunate, Tokugawa Hidetada changed the title of king to taikun (大君), as a sign of respect to the Japanese emperor. Thereafter, Tokugawa Ienobu switched the title back to king, only to be changed once again to taikun by Tokugawa Yoshimune.
In 933, Taejo of Goryeo was conferred the title of King of Goryeo (高麗國王) by the Emperor Mingzong of Later Tang. Prior to its capitulation to the Yuan dynasty, imperial designations and terminology were widely used by Goryeo domestically. Its rulers claimed to be the Son of Heaven, as did Chinese emperors. Gyeongsun of Silla addressed Taejo of Goryeo as the Son of Heaven when he capitulated. Even though the Song dynasty, Liao dynasty and Jin dynasty were well-informed of Goryeo's use of imperial titles, all three Chinese dynasties tolerated such practice. In 1270, Goryeo became a semi-autonomous region of the Mongol Empire, bringing an end to its imperial system. Its rulers bore the title king and were prohibited from having temple names which were reserved specifically for emperors. Gongmin of Goryeo declared independence from the Yuan dynasty in 1356.
In 1392, Taejo of Joseon overthrew Goryeo and established a new dynasty on the Korean Peninsula. He was bestowed the title King of Joseon (朝鮮國王) by the Hongwu Emperor of the Ming dynasty. Both domestically and externally, rulers of Joseon held the title of king, unlike Goryeo's domestic use of imperial title.
Lê Hoàn abolished Đinh Phế Đế, the son of Đinh Bộ Lĩnh, and was bestowed the title of Tĩnh Hải quân Jiedushi in 986, when the emissary of Song has visited. In 988, Lê Hoàn was promoted to proxy Grand Commandant (檢校太尉); in 993 to Prince of Jiaozhi (交趾郡王) and finally in 997 the title as King of Nanping (南平王).  Nanping meaning Southern Pacific.
In 1010, Lý Thái Tổ established Lý dynasty and granted the title of Prince of Jiaozhi (交趾郡王) by Song Emperor Zhenzong. In 1174, Lý Anh Tông was bestowed the title King of Annam (安南國王); "Annam", meaning "the Pacified South", was the name of Vietnam during Chinese rule. Domestically, Vietnamese monarchs maintained the title of emperor.
During the Trần dynasty, Vietnam successfully resisted three invasions by the Mongol Empire and the Yuan dynasty. Thereafter, the Ming dynasty administered Vietnam for two decades. Vietnamese soon came to see their state as the "southern state" in relation to China as the "northern state".
During the Lê dynasty, after defeating the Ming, Lê Thái Tổ claimed kingship with the title Đại Vương (大王). It wasn't until Lê Thánh Tông did Vietnamese rulers reclaimed the title of emperor. The system continued to be used until the end of the dynasty itself, as all rulers claimed the status of emperor domestically and referred to as king when dealing with China.
During the Nguyễn dynasty, Gia Long was conferred the title King of Việt Nam (越南國王) by the Jiaqing Emperor of the Qing dynasty. While the Nguyễn dynasty accepted Chinese suzerainty and adopted the title of king when dealing with China, . Domestically, Nguyễn monarchs also used the title emperor and called its realm the "southern dynasty" (vis-à-vis Qing dynasty the "northern dynasty"), implying an equal status with China.
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