half asleep and half awake; trance
Describes a state between sleeping and waking, or a dreamy, trance-like condition. Often used in the phrase 夢うつつで.
I was so tired that I listened to the lecture half asleep.
彼女は夢うつつの状態で、名前を呼ばれても気づかなかった。
She was in a trance-like state and didn't notice even when her name was called.
dream and reality
Refers to the boundary or mixture of dream and reality. Often used in expressions like 夢うつつの境 or 夢かうつつか.
夢うつつの境がわからなかった。
I could not tell the boundary between dream and reality.
夢うつつの境をさまようような気分だった。
I felt as if I were wandering the border between dream and reality.