  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
that the issue of explicit CP invariance can be linked to the existence of a
basis in which all coefficients are real; with even a theorem proposed a decade
ago claiming that the scalar sector of any multi-Higgs doublet model is
explicitly CP-conserving if and only if all of its coefficients can be made
real by a basis change. This is compounded by the fact that in all specific
multi Higgs models considered so far, the calculations complied with this
claim. Here, we present the first counterexample to this statement: a
CP-conserving three-Higgs-doublet model for which no real basis exists. We
outline the phenomenological consequences of this model, and notice that the
extra neutral Higgs bosons are neither CP-even nor CP-odd but are "half-odd"
under the generalized CP-symmetry of the model.
