The Japanese use 'mai-hoomu' and 'mai-waifu' (my-home and my-wife) as compound words in Japanese.

I am not sure if this is quite what you mean, but it is significant in that generally, in Japanese, you wouldn't use a possessive. It would be understood that you were talking about your own home. So there is no 'native' Japanese equivalent for these words.

Also, in Japanese they carry a significance that they wouldn't have in everyday English. 'Mai-hoomu' can be ironic, or can have a very strong flavour of real estate agent-ese. It's what you would use in an ad to convey the glorious dream of owning your own home.

So they are new Japanese words, invented from the English but with new meanings.

Just at the moment I can't think of another Japanese-coined 'non-native Japanese' word used in everyday English, but I'll put my thinking cap on!