In models with several fields with identical quantum numbers, there exists large freedom of defining the CP transformation. Different choices are possible and can have consequences which defy the common wisdom. Here, I will describe a CP-conserving three-Higgs-doublet model based on an order-4 generalized CP symmetry, in which it is impossible to make all coefficients in the scalar potential real. This model is a counterexample to the claim, published a decade ago, that the existence of real basis and explicit CP invariance are equivalent for any multi-Higgs-doublet model. Another peculiar feature of our model is that it contains neutral physical scalars which are neither CP-even not CP-odd but are CP-half-odd.