Well, all of the above, for different reasons.
Our Great Grandmaster Dr Chi Chiang-Tao, and his teacher Cheng Man-Ching used the older style of Romanisation for spelling their names and that of their art in English, so who are we to change it? Several of our lineage masters were expelled from China or fled to Taiwan and the West during the Cultural Revolution, when T’ai Chi, Traditional Classical Chinese medicine, Buddhism, Taoism, chi kung and so many more treasures of traditional Chinese culture were banned, supressed or ridiculed as ‘backward’ by the new government. Out of respect for their lives, choices and journeys to bring T’ai Chi to thousands of people all over the world, we spell their names and art as our Masters did: T’ai Chi Ch’uan. (The hyphens and umlaut on that from 1950s phonetics have long since passed into obscurity).
Interestingly, T’ai Chi was only officially allowed to be practiced again in mainland Communist China in 1974 when Chairman Mao mentioned it briefly in a speech, after decades in the shadows. This story is movingly written in a chapter in ‘Chen Taichi’. When I find the book’s details again I will post them here. Meanwhile T’ai Chi had blossomed in Taiwan, Hong Kong and all over the world where it was appreciated as a peerless martial art, moving meditation and way of health.
Pinyin is the modern official Chinese government preferred method of Romanisation and is widely used digitally including on smart phones, so it has become the standard. In this system one writes: taiji quan.
Also common is the halfway house of ‘taichi‘, really common in western countries. That’s fine too, and I use that interchangeably, especially when typing an email quickly…
All of them should really be pronounced something like ‘tie – ji – juen’, but often get Anglicised as ‘tie – chee – chew – ahn’ (which comes out sounding like ‘big asthma breathing’ to a native Chinese speaker…) But never mind, generally, if you say or write any of these things, then most folks will understand what you are saying or writing. One could get very serious about it, but think of words like ‘restaurant’ and how we in the UK garble it and add a hard ‘T’ at the end, and you get the picture.
Tomorrow I will write about how ‘T’ai Chi Ch’uan’ as the title of our martial art translate (or fail to translate) into English.