Etymology: Contrary to popular belief (google the term for examples), this is in fact a word, and can be cited as far back as the 17th Century:
1675 R. Burthogge Cavsa Dei 352 'A many false, supposititious, and ingenuine' Writings.
Source: Oxford English Dictionary online.
The word can be made by attaching the prefix in- (a variant of un-) to the existing word 'genuine', in parallel to in- + sincere, in- + compatible, in- + valid and so on.
Although it may often be confused with 'ingenuous', this is a simple malapropism and does not affect the validity of the word.
2. Her smile was ingenuine.
3. The tourist attraction gives an ingenuine experience of life in the 1500s.