Emily Dickinson wrote over 1700 poems during her lifetime. It would be impossible to define her Romantic or Realistic characteristics in all of her poetry. That said, I will offer multiple examples of poems which illustrate her use of each.
Romantics tended to focus upon nature and imagination. They found peace in the countryside and valued intuition over reason. Dreams and personal emotions were important. Poems which illustrate characteristics typical of the Romantic period are "Nature, the Gentlest Mother," "Two Butterflies Went Out at Noon," and "A Narrow Fellow in the Grass."
"Nature, the Gentlest Mother" includes very vivid imagery regarding nature. The speaker tells of rampant squirrels, impetuous birds, and fair conversations as the sun sets. "Two Butterflies Went Out at Noon" depicts butterflies dancing above a stream. They go on to fly over the ocean and never be seen at any port. Their disappearance remains a mystery to the speaker. Lastly, "A Narrow Fellow in the Grass" is a poem about a snake. The grass splits as the snake slithers through, commanding nature in its own way. The snake's presence takes the speaker's breath away.
Realism depicts life as it really is. Realists require true to life settings, characters, and situations. They act as observers, refusing to intervene on behalf of the characters (in order to make things easier). Poems which illustrate Dickinson's Realism are "The Soul Selects Her Own Society," "I'm Nobody," and "I Can Wade Grief."
"The Soul Selects Her Own Society" illustrates the idea that life is what it is. Unmoved by either emperors or chariots, the "her" of the poem is stone. Nothing else matters. In "I'm Nobody," the speaker sets up an immediate paradox. This illustrates the general confusions in life (things which seem impossible can be possible). This is a realism in life. Similar to "I'm Nobody," "I Can Wade Grief" speaks the to reality of life. Grief exists; "power is only pain"; and weights (troubles, trials, and tribulations) will hang from us.