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.