A Google Books search indicates that use of "real cool" as an intensifying phrase along the lines of "genuinely [or very] cool" goes back (in colloquial written usage) to at least 1833. From Duty D. Doubikins, "Of Malting Indian Corn, and a Pretty Considerable Lot of Things Besides," in Mechanics Magazine and Journal of Science, Arts, and Manufactures (July 27, 1833):
I shan't tax ye more than a dollar and quarter per week for your board, and daughter Becky keeps the house awful clean, though we an't got much help for that part; and you'll have for breakfast boiled pork, and roast pork, and molasses that'll stick your ribs together, and tea and coffee, and oceans of milk—Becky always 'livers the milk herself, because we are short of help—and bread and butter, better nor the best Goshen,—our dog Watch always churns the butter in the cellar, a real cool one I tell ye; but that Watch he's a cruel cute critter, he does'nt like churning; and a prime churn it is, all my own invention, and something like your treadmills, only the wheel's a 'clined plane instead; you shall see it, Mister, when you come.
Similarly, from Laurie Loring Pratt, The Holiday Album for Girls (1875):
“It's too hot!"
“O! it's real cool out under the trees,” urged Lottie.
“I can't go till Carrie comes. There is no fun walking with little girls;” and Kate turned away, taking no more notice of her little sister.
From John Habberton, Four Irrepressibles; or, The Tribe of Benjamin (1877):
"Ben would like the Episcopal Church, wouldn't he, Aunt Agnes?" continued Rob. "I kind of like it too; hopping up and down rests a fellow a lot, and the minister looks real cool and nice with that white dress on."
And from a letter from a six-year-old girl in Harper's Young People (July 11, 1882):
I have a blackboard: I print, and can add and take away. I am in the Second Reader. Mamma and I are going to Maine next month to stay till it is real cool here. There we go out fishing. We pick blueberries. blackberries, and cranberries. I have four little cousins who go from here. We all have the same grandpa and grandma.
And finally (from a much later date), from "The Manager's Page," in To-day's Cinema News and Property Gazette (September 10, 1913):
Why not give to your attendants light, comfortable, white duck uniforms? Make them look clean, neat, and cool; you will then create the impression that your place is a real cool place, a good place to rest and escape the boiling sun rays.
These examples show increasingly wide usage of the phrase in writing by the 1880s. A search of the Library of Congress's Chronicling America database of newspapers has a first occurrence of "real cool" in the Washington, D.C., Evening Star (October 28, 1858):
The last three mornings the weather has been real cool, decidedly winterish, considerable ice forming in many places.
Eight or nine additional unique occurrences of the phrase in the sense of "genuinely cool" (and not "genuine cool") appear in the Chronicling America archives from the 1870s.
Though you wouldn't expect Henry James to have one of his upper-class narrators say "real cool" when he (or she) meant "genuinely cool," the expression has appeared in published writing for more than 180 years, and probably has been a part of spoken English for much longer. Whether it is correct is, in my view, a cultural style question and (therefore) has no absolute yes or no answer.