Have you ever experienced the opposite of lucid dreaming?

If lucid dreaming is waking up inside a dream and being aware you're dreaming, as well as being able to control that dream. I myself have been experiencing the exact opposite. I will be asleep, dreaming whatever thing it is I dream and then I'll wake up, the dream will be there in the waking world, I just continue the dream story, turn the pillow over or whatever it is I do, go back to sleep and the dream will have persisted. I love this, never had it before. Even if the period of being awake is a bit longer, the dream remains and I can just pick it back up.

Does this sound familiar to any of you or am I even stranger than I thought?