Type-Z models, where charged leptons, up type quarks and down type quarks each couple to a different scalar, are only possible when there are three or more Higgs doublets. We consider the Type-Z three Higgs doublet model imposed by a softly broken Z 3 symmetry. We take into account all theoretical and experimental constraints, highlighting the importance of perturbative unitarity and bounded from below conditions that we develop here, and which have a strong impact on B → X s γ. Since there can be cancellations between the two charged Higgs in B → X s γ (and in h → γγ), the lower bounds obtained on the charged Higgs masses are alleviated. We also discuss in detail the important physical differences between exact alignment and approximate alignment.