  We study the possible existence of a real Cabibbo-Kobayashi-Maskawa (CKM)
matrix, with CP violation originating from physics beyond the standard model
(SM). We show that present experimental data allow for a real CKM matrix
provided that new physics also contributes to Delta m_{B_d} by at least 20% of
the SM contribution (for rho > 0), besides generating CP violation in the kaon
sector. The naturalness of a real CKM matrix is studied within the framework of
general multi-Higgs-doublet models with spontaneous CP violation. As an
example, we discuss a specific two-Higgs-doublet model and its implications for
CP asymmetries in non-leptonic neutral B-meson decays.
