In a dreamy, distracted, or drowsy manner.
From dreamy + -ily (adverbial suffix from Old English -ile). This creates an adverb expressing the manner in which something is done, similar to how 'happily' comes from 'happy.'
The -ily adverb suffix is fascinating because it often requires dropping the 'y' before adding the suffix—'dreamy' becomes 'dreamily'—a pattern that trips up spellers but has roots in Old English morphology.
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