(This probably belongs a bit deeper in the task hierarchy under more general advanced / power user search, but I can't find a ticket for that.)
While phonetic or "sounds like" search is typically very course-grained, and can often be (expensively) emulated with a regex, it could still be a useful feature for power users. The prototypical use is casting a wide net for matching names.
Elasticsearch already supports major phonetic searching algorithms, and their enthusiasm-crushing description on that page is actually quite fair. Phonetic search algorithms can be very language-specific (esp. since many are developed for English and English spelling is a travesty and a tragedy).
We could also support more than one type of phonetic searching, especially for transliteration. @santhosh pointed me to an interesting library, "Indic Soundex" (description, GitHub) for cross-script/cross-language searching in Indic languages.
We can talk about use cases (searching names, searching transliterations, lack of keyboard availability, etc.), implementation details (new indexes, slow regexes, a custom plugin), which algorithms to support, and how to make them available to searchers.