Meltype
https://github.com/yksr-melt/Meltype
This is an experimental IME that lets you type a mix of English and Japanese without switching between the two. It should be smart enough to identify the language of substrings.
kyouhagoogledekensaku → 今日はgoogleで検索
Unfortunately I haven’t been able to test it because it’s developed primarily for Windows and the Mac version seems pretty buggy, but it’s a great idea.