Many people who first see a high dynamic range (HDR) display get the impression that it is a 3D display, even though it does not produce any binocular depth cues. Possible explanations of this effect include contrast-based depth induction and the increased realism due to the high brightness and contrast that makes an HDR display "like looking through a window". In this paper we test both of these hypotheses by comparing the HDR depth illusion to real binocular depth cues using a carefully calibrated HDR stereoscope. We confirm that contrast-based depth induction exists, but it is a vanishingly weak depth cue compared to binocular depth cues. We also demonstrate that for some observers, the increased contrast of HDR displays indeed increases the realism. However, it is highly observer-dependent whether reduced, physically correct, or exaggerated contrast is perceived as most realistic, even in the presence of the real-world reference scene. Similarly, observers differ in whether reduced, physically correct, or exaggerated stereo 3D is perceived as more realistic. To accommodate the binocular depth perception and realism concept of most observers, display technologies must offer both HDR contrast and stereo personalization.