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The acronym "TARDIS" shall be rendered unitalicised and in all uppercase when it refers generally to the time-travel devices invented by the Time Lords, as well was when it denotes the Doctor's TARDIS specifically.
"Tardis" shall refer to the name of this wiki, as is obviously implied by the project namespace, Tardis:
Yes, the first 30 years of Doctor Who tie-in material, not to mention modern day dictionaries, actively prefer the style "Tardis", but it's necessary to maintain a strict distinction between the in-universe objects and the name of the wiki.
Despite the predilection of 1960s Doctor Who fiction to italicise Tardis as if it were the name of a unique ship, please don't italicise it on this wiki. TARDIS is the name of a type of object, as commonplace as "airplane" and "car" are in the real world. There are many, many TARDISes out there, so it would be incorrect by most English language standards to italicise the name in most situations.