This is an excellent question, because it gets right to the heart of the problem of mathematical realism which is essentially the same question as the problem of universals, which is essentially the same as the mind-body problem of Cartesian dualism.
At the heart of all of these questions is the issue of do we live a purely material existence, or is there a Ghost in the Machine.
So while I know nothing much about mathematical realists such as Roger Penrose, nor the reasons why they believe as they do, I can say how I think these issues impact what I think mathematical realism ought to be.
The issue in @Alexander’s question boils down to one of complexity. It seems simple enough to see the implied order of the integers, and then intuit that these are real for everyone therefore must have independent reality. But when it comes to complicated and inventive solutions to problems that maybe are not necessarily problems demanding resolution in the first place, then it intuitively seems that such creativity has to be original in all ways.
So the question is by implication about creativity, about Art, and where if anywhere we draw a line between Art and Science,
One way to address the issue is to consider poetry and the dictionary. The dictionary by its very existence implies all possible arrangements of the words it contains; therein lies its ghost, and among those possibilities are the complete works of Shakespeare, which we theoretically believe could be replicated by an army of chimps banging away on typewriters. So when Shakespeare writes “What's in a name? That which we call a rose by any other word would smell as sweet;” the phrase [and its words] gains meaning from its context in the play, just as the play gains meaning by being brought into the world by Shakespeare, and then more so from performances, its place in history, and the impact it has had on countless lives, and the spin offs in music and film.
All of this contextual meaning amplifies parts of the ghost that are in the potential in the dictionary, and therein lies the Art, the creative process of taking something of little meaning and imbuing it with greater meaning. This is what Marcel Duchamp did with Fountain, as did his audience.[Though not the exhibition panel!]
Creativity in mathematics is no different to creativity in art, literature, or any other act in life. It multiplies the sparse meaning in the ghostly relationships between universals, by giving life to them in ways that are to a greater or lesser extent original, which is borne out by the value we place on originality and creativity.
The bottom line is that far more complex and unpredictable solutions are added to the ghost by all of life, so there is no specific problem with mathematical complexity The problem is with the original premise of a ghostly mathematical realism in the first place. Is it really independently real?