I don't think so in real sense. [Technically correct, but you would never hear it.] I don't think so in the real sense. [Correct, but seldom expressed this way. You would more likely hear it as "...in the real sense of the word/phrase.", where the word or phrase has been mentioned already in the context.] I don't think so in a real sense. [Correct and the most often form heard.] Oksana gives an excellent quote and example.

Quote by Martin Luther King Jr. : “In a real sense all life is inter-related. All men are caught in an inescapable network of mutuality, tied in a single garment of destiny. Whatever affects one directly, affects all indirectly. I can never be what I ought to be until you are what you ought to be, and you can never be what you ought to be until I am what I ought to be... This is the inter-related structure of reality.”