You could split the stream into N chunks. Every time you watch the movie, M of them are selected and fudged just a little bit. Since M << N, this can be done in real time. The next user sees the same chunks you saw, except that some new chunks have been fudged also.
Thus the movie slowly drifts away from the original. To counteract this, fudges are more likely to carry the chunk towards the original then away from it.
This is what I came up with in 5 min, so there might be problems with the approach.