Why is anime not spelt phonetically?

Solution 1:

Anime is the shorted form of animēshon, which is just the Romaji form (i.e., Japanese written in Roman letters) of the word "animation", which the Japanese adopted into their language with approximately English pronunciation. Check here.

Solution 2:

In a comment, John Lawler wrote:

It is a phonetic spelling. It reproduces the sound of the Japanese word アニメ /a ni me/ in the Katakana syllabary. English spelling does not represent sounds at all well; it's so poor (it was designed for Middle English, after all) that nobody ever uses it when accurate phonetic transcription is needed. For that one uses Modern English phonemic symbols. In that phonetic spelling, the word would be /anime/. English spelling is not ready for prime time.