Tone sandhi is the process by which the nominal tones of syllables or words change based on the surrounding context.
I know that Mandarin Chinese and Thai have tone sandhi - but is this process universal within tonal languages?
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"Tone sandhi" is a vague terms so without a definition of what it is, I don't see any way to know. The term is usually used to refer to phonological changes of category that apply at the phrasal level. Every phonological object is subject to some contextual micro-adjustment (e.g. coarticulation), but not all such processes are fully phonological.
The Chadic language Angas is one language that has no phrase-level categorial modifications of tone as a function of preceding or following tone within the sentence. Tone sandhi is not universal. It is extremely common, however: it is possible that there are such rules in the majority of languages that have tone, though Xiamen-style tone replacements are rare. Under-reporting is a standard problem in the literature. Under-analysis is another problem. One might claim that there is some strange tone sandhi in Angas involving nouns plus certain modifiers, but actually there are some floating tone suffixes that function as "linkers" or case markers. I might be inclined to say that almost all languages have some phrase-level tonology, but that is heavily influenced by Bantu.
The most common form of "tone sandhi", in my experience, is that a tone rule which applies within a word may apply between words as well, that is tone often has a larger-than-word domain. This is as opposed to vowel harmony, which strongly tends to be word-limited. The second runner-up pattern is "applies / doesn't apply pre-pausally", for instance spreading of H to the right does not apply to a pre-pausal target (many languages).