Short answer: The last section seems "real" as it describes the feelings of Peyton Faquhar.
Ambrose Bierce's "An Occurrence at Owl Creek Bridge" is a story told from three separate points of view that serve to cause the reader wonderment about what is real and what is not. These three vantage points are (1) omniscient, in which the narrator is aware of everything about the characters and events; (2) objective, in which the narrator acts much as a camera, reporting without comment; and (3) third-person limited, in which the narrator gives close focus to the thoughts and feelings of one character.
The section that seems the most real to the reader is, ironically, the one that is, in fact, not real. But, because it records the feelings and last thoughts of the main character, Faquhar Peyton, the reader feels drawn into the narrative with Bierce's use of strong sensory images, the use of the third-person limited and, at times, first person narrative. In this third section Farquhar's senses have sharpened and he senses pain as his hands thrust the rope from his neck from his neck.
...his neck ached horribly; his brain was on fire; his heart, which had been fluttering faintly, gave a leap, trying to force itself out of his mouth. His whole body was racked and wrenched with an insupportable anguish!
Further, he experiences the small details of nature with an unusual intensity. And, all the sensations that he feels are so graphic and detailed that the reader is captivated and believing, drawn into the illusion. Nevertheless, the story is real because, although the order of what is described is illusionary, it is only because of the order of what is described; the incidents are real and the sensations of Faquhar are real. For, his experience of having an aching neck and sense of being spun around by the stream and then, finally, his incapability to feel his weary feet certainly could just as well be the feeling of an exhausted man who has escaped as those of a hanged man.