Real is an adjective. It only modifies nouns or pronouns.

Really is an adverb. It modifies verbs, adjectives, or other adverbs.

Correct: He stayed at hotels with real class.
(Class is a noun. The adjective modifies it.)

Incorrect: He stayed at a real classy hotel.
(Classy is an adjective. It should be modified by an adverb.)

Correct: He stayed at a really classy hotel.
(The adjective classy is modified by the adverb.)

See also Sure or Surely?

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