SPOILERS: I'm sure when people re-examine it there may be things that can be interpreted as clues but I also think DiCaprio dreamed the whole thing. Some of the early parts of the movie made little sense and his wife refers to them as mysterious goons from shadowy organisations so it may be that the strange vagueness of the early scenes was deliberate. The fact that the spinning widget was his wife's and not his is a clue for me, although it can be interpreted any way. If his wife was French and her parents were French and English, his children could have dual nationality so it seems strange that they had to be stuck in the USA forever.
It was hugely entertaining. I wish the makers of the modern Mission Impossible movies could have produced something with more of a vibe like this movie.