Ambrose Bierce’s short story “An Occurrence at Owl Creek Bridge” might be considered a reflection of the “realist” movement in American literature for a number of reasons, including the following:

• In its opening section, it tries to report facts as accurately and unemotionally as possible. Its opening sentences, for instance, are perfect examples of this neutral, objective, “realistic” style of writing:

A man stood upon a railroad bridge in northern Alabama, looking down into the swift water twenty feet below. The man's hands were behind his back, the wrists bound with a cord. A rope closely encircled his neck. It was attached to a stout cross-timber above his head and the slack fell to the level of his knees.

Much of the language of the story – particularly in the first section – resembles the language just quoted. It is crammed with information, dry in tone, and implies a narrator who is highly reliable.

• Even when the narrator engages in speculation (as in the sentence “Doubtless there was an outpost further along”), his speculations seem reasonable and realistic.
• The narrator seems to be highly concerned not just with facts in general but with military facts in particular. He seems familiar with the details of military procedures and military life, as when he mentions  

a single company of infantry in line, at "parade rest," the butts of the rifles on the ground, the barrels inclining slightly backward against the right shoulder, the hands crossed upon the stock.

• The narrator makes few evaluative comments, especially in the opening section.  One exception to this rule is the sentence beginning “Death is a dignitary,” but in general the narrator seems to feel no great need to express his own opinions. He basically sticks to the facts, or at least seems to.
• When the narrator speculates, he calls explicit attention to this fact, thus (paradoxically) increasing our trust in him.  Thus, at one point he says,

The man who was engaged in being hanged was apparently about thirty-five years of age. He was a civilian, if one might judge from his habit, which was that of a planter. [emphasis added]

• By emphasizing dry, realistic exposition in the beginning of the story, Bierce creates a certain amount of suspense (we wonder: why all these details?), and also enhances, through contrast, the exciting plunge into stream of consciousness later in the tale.
• When the story finally does enter the stream of Farquhar’s consciousness, even much of the phrasing there seems designed to be precisely realistic and credible, at least at first. Only later – at the very end of the tale – do we realize that all this apparent realism has been the figment of the imagination of a desperate, dying man.