Can "real" be used as an adverb to describe an adjective?

The Longman Dictionary of Contemporary English lists 'real' as an adverb also, but qualifies it as 'American English spoken'

Even Merriam-Webster's Online Dictionary has the following note for 'real as an adverb:

Most handbooks consider the adverb real to be informal and more suitable to speech than writing. Our evidence shows these observations to be true in the main, but real is becoming more common in writing of an informal, conversational style. It is used as an intensifier only and is not interchangeable with really except in that use.

I, therefore, don't think it's incorrect to say something like "It is a real cool answer" in informal speech and writing! Here, 'real' is an intensifier, that is, an adverb qualifying an adjective!

"real cool" might be incorrect, but is used in casual conversation.

Even if "really" is "more correct", the Common Errors site mentions:

“Really” is a feeble qualifier.
“Wonderful” is an acceptable substitute for “really great” and you can give a definite upscale slant to your speech by adopting the British “really quite wonderful”.

Usually, however, it is better to replace the expression altogether with something more precise: “almost seven feet tall” is better than “really tall”.
To strive for intensity by repeating “really” as in “that dessert you made was really, really good” demonstrates an impoverished vocabulary.

I assume when you say "correct" you mean "standard". And I will presume that when you don't say "standard", you mean US standard English. In which case, the answer is that this is not standard US English. It's common in certain dialects, I believe mainly southern dialects. There's nothing wrong with it, it's just not standard.