Where you have a number of studies with conflicting results, it is time for a meta-analysis, which carefully combines a number of studies, to get a more statistically powerful sample.
This study was cited by a meta-analysis:
G. James Rubin, Jayati Das Munshi and Simon Wessely, Electromagnetic Hypersensitivity: A Systematic Review of Provocation Studies
Psychosomatic Medicine 67:224-232 (2005), doi: 10.1097/01.psy.0000155664.13300.64
Their abstract covers the territory very well:
Objectives: The objectives of this study were to assess whether people who report hypersensitivity to weak electromagnetic fields (EMFs) are better at detecting EMF under blind or double-blind conditions than nonhypersensitive individuals, and to test whether they respond to the presence of EMF with increased symptom reporting.
Perfect - that seems to be directly aimed at your question.
Results: Thirty-one experiments testing 725 "electromagnetically hypersensitive" participants were identified. Twenty-four of these found no evidence to support the existence of a biophysical hypersensitivity, whereas 7 reported some supporting evidence. For 2 of these 7, the same research groups subsequently tried and failed to replicate their findings. In 3 more, the positive results appear to be statistical artefacts. The final 2 studies gave mutually incompatible results. Our metaanalyses found no evidence of an improved ability to detect EMF in "hypersensitive" participants.
I'm not sure where the Real et al study fit in here, but if it gave a positive result, it was in the minority. (Although, it should also be considered by its sample size and methodological strength, which aren't covered by the numbers here.)
Conclusions: The symptoms described by "electromagnetic hypersensitivity" sufferers can be severe and are sometimes disabling. However, it has proved difficult to show under blind conditions that exposure to EMF can trigger these symptoms. This suggests that "electromagnetic hypersensitivity" is unrelated to the presence of EMF, although more research into this phenomenon is required.
So, nocebo appears to be a plausible explanation.