timepiececlock

I think I do a mix of that. Certain stuff, like "Evangelion", I have to pronounce as if they were English because I can see/hear the English root word, or in this case it simply *is* an English word, with somewhat peculiar conjugation. It strikes me as silly to pronounce an English word with a Japanese mispronunciation simply because it's used in the title of a Japanese show/series/manga. The Japanese people who chose to use that word know it's not Japanese, I know it's not Japanese, and the viewers/readers know it's not Japanese. On the other hand, I've heard people give English words like that a Japanese pronunciation because they want to distinguish it as being specific to that manga or anime series as opposed to common use of the word.

For actual Japanese words/names/titles, for me it seems to depend on how I get introduced to it and how easy it is to "say". I find "ann-ih-may" easier and swifter to say aloud than "ahn-ee-may". On the other hand, I learned the word "manga" as "mahn-gah" not "mang-a", so that's how I pronounce it. Naruto I pronounce based on what I've actually heard in the subs, which means any of my listening biases that filter sounds probably affected how I pronounce that.