"real (or mathematically viable)"

We cannot say for sure that such manifolds -- the non-Kähler ones -- are even real (or mathematically viable).  There is no broad existence proof, such as that pertaining to Calabi-Yau manifolds, and existence, so far, has only been established in a few isolated cases.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 243

(I quoted that passage for its nice mathcentric gloss “real (or mathematically viable) “, but the paragraph is actually paradoxical.  Up to the word “manifolds”, it makes sense and should mean: non-Kähler manifolds have been defined in principle, but so far we have seen no examples; not “mathematically viable” would mean that no object could actually meet the definitional criteria, hence we shall never come across any.  But then he adds, that some finite number of instances  have indeed been proved to exist (albeit perhaps only in the shadowy form that you get from an “existence proof”).

For essays on the subject “real in mathematics” (in the ontological sense, not in the retronymic sense of “not containing the square root of negative one as a factor”), try these:

https://worldofdrjustice.blogspot.com/search/label/Platonism