"You eat light."
the word does exist, it was used in my French classes, which I took for 12 years. Be careful about claiming what words do or do not exist, especially if you aren't a native speaker because there isn't just ONE French...any language is a combination of dialects and chronology in which new words get invented all the time and used in different ways. A better strategy is to understand the elements of the word and figure out the meaning from there: "léger" = 'light', "-ment" = 'adverbial ending (english -ly)'.
I think the 'light' here doesn't modify how one eats (making it an adverb), but what one eats, which would make it an adjective. I've had this same discussion about why we say "pack light" because it sounds so wrong initially, but after some thinking I don't think I would use it to describe the actual packing, but rather the result. I guess I'm getting a bit off topic from the whole French thing, but what do you make of that?