CP-conserving multi-Higgs model with irremovable complex coefficients

Models beyond the Standard Model (bSM) often involve elaborate Higgs sectors, which can be a source of CP violation. It brings up the question of recognizing in an efficient way whether a model is CP violating. There is a diffuse belief corroborated by all specific multi-Higgs models considered so far that the issue of explicit CP invariance can be linked to the existence of a basis in which all coefficients are real. Here, we prove that this belief is not justified. We present a CP-conserving three-Higgs-doublet model for which no real basis exists. The generalized CP symmetry of this model is of order 4, which leads to the peculiar property of the extra neutral Higgs bosons of being neither CP-even nor CP-odd but “half-odd” under this symmetry transformation.