  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 $B \rightarrow X_s \gamma$. Since there can be cancellations between
the two charged Higgs in $B \rightarrow X_s \gamma$ (and in $h \rightarrow
\gamma \gamma$), 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, and present some useful
benchmark points.
