AHQ: How would Chinese Emperors address foreign leaders outside of Asia

(This is a weird one, as it's more of a question + Alternate history scenarios)

The Chinese Imperial Tributary System is a complicated one. For thousands of years, China was the hegemonic power in Asia, with various countries such as Korea, Japan, Vietnam, Thailand, Cambodia, Central Asia, at one point were tributaries to the Chinese.

While it can be seen vassals submitting to their hegemon, that wasn't really the case. Instead, it was more in exchange for acknowledging Chinese supremacy, you'll get special privileges, such as mercantile ties, gifts, and even political legitimacy and protection from the Chinese. However, the most interesting part of the concept is how the Chinese Emperors addressed other foreign rulers. Since there could only be one Son of Heaven, that meant other nations had to be subordinate to the one Emperor, meaning a lot of them addressed themselves as 'Kings' most of the time. For example, Chinese Dynasties called the Emperors of Japan instead "Kings of Japan," and did so later for the Ashikaga and Tokugawa Shoguns.

This led to the concept of 'Emperor at Home, King abroad," where leader addressed themselves as 'kings' when dealing with the Chinese, but domestically has a title equivalent to an 'emperor.' This can be seen in beforementioned Japan, and in Vietnam where the Chinese addressed their leaders as 'Kings' but internally used Imperial titles.

So, why am I mentioning all this? My question is how the Chinese Emperor address would foreign powers such as in Europe, Africa, India or the Middle East? IOTL, a letter from the Qianlong Emperor simply addressed George III as 'King,' which was already his title back at home. Another is that they called the Roman Empire itself 'Daqin.' (Though, I can't find how they'd address the Roman Emperor) So, what if the Chinese Emperors had more contact with foreign leaders, and what would they call them?

For example, how would the Chinese Emperor address, say, the 'Emperor of France' as, or the 'Holy Roman Emperor' or 'Ottoman Sultan,' 'Mughal Emperor, 'Roman Emperor,' 'Shah of Iran' or 'Tsar of Russia'. And if we're getting more outlandish with history, how would they address titles such as 'Chieftan,' 'Emir,' 'Tlatoani,' or even more lesser titles such as 'Dukes' and 'Princes,' and even 'President'

Would they Chinese Emperors simply and mundanely address them by their usually titles (Whether it'd be King or Emperor)? Or would they just try to come up with a 'King' title in lieu of their usually one used back home? What would be the Chinese convention for this?
 
Probably they with some way had some title for emperors of Russia, Germany and Austria-Hungary since they co-existed with Qing Dynasty.
 
The Qing Emperors, in letters, addressed the Presidents of France and the United States as 大伯理玺天德 (meaning The Great Seal of Heaven), so there's that.
大伯理玺天德 (Great Bolixitiande) It's actually an achaic direct phoenetic translation of "president" to Chinese, the characters used were only meant to represent the sound, not the literal meanings.
 
大伯理玺天德 (Great Bolixitiande) It's actually an achaic direct phoenetic translation of "president" to Chinese, the characters used were only meant to represent the sound, not the literal meanings.
Though the question is how would they have styled the leaders of republics in a world where, say, Consul or Stadtholder ended up the favored title for the leader of a Republic.
 
Though the question is how would they have styled the leaders of republics in a world where, say, Consul or Stadtholder ended up the favored title for the leader of a Republic.
困苏二 kunsu’er (kuhn-soo-er)
私太都的二 sitaidoude’er (ss-tie-doh-der)
 
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