Physical theories are mathematical models that approximate reality. If the models of any science were real and not representations, or even if they were accurate representations, that science as a whole would simply be finished with its job. The fields of the field equations aren't real and neither are the mathematical tensor spaces the operate on. Space is real, and they are abstract models of how it behaves.
So being used by physics does not make math real. On the other hand, from an intuitionistic point of view (a la Kleene), mathematical models are as real as your vision or your emotions are.
Mathematics and logic are patterns of communication to which humans naturally respond. Euclidean space has the same level and kind of reality that fear does. Is fear real? Well, that depends... But however you handle fear ontologically, math goes in the same bucket. And intuitionism affords that bucket the status of reality. It is real as a set of patterns, and those patterns can be empirically validated or falsified by exposing them to humans.
Four-dimensional space has the same level of reality as other patterns of reaction to other stimuli, say, the fear of spiders. If you are one of the people who share and understand it, it produces a given feeling. In the arachnophobe's case, anxiety of a common form, in the mathematician's, a sense of settledness that is common to most mathematical propositions.