Gaijinese can be Japanese spoken with a strange anglicized accent or Japanese that's slightly off due to mistakes in grammar (Japanese spoken with English or another language's grammatical structures), otherwise known as "hen na nihongo" (strange Japanese).
Gaijinese could also be English spoken with inclusion of many Japanese words; for example, "Oh, that looks so abunai dangerous", "hahah! he is so omoshiroi interesting/funny", etc.
Huh? What are you saying? That's not really Japanese, man! It's Gaijinese!