  The $\mathbb Z_2\times \mathbb Z_2$-symmetric 3HDM potential, sometimes
called Weinberg's 3HDM potential, can accommodate both explicit and spontaneous
CP violation as well as natural flavour conservation and is hence relevant for
studies of CP violation coming from a realistic extended Higgs sector. The
model has an interesting regime, controlled by a single parameter, where it
acquires an approximate global $U(1) \times U(1)$ symmetry which causes CP
violating effects to be small and two pseudo-Goldstone bosons to be present in
the scalar spectrum. From parameter space scans, we find that, in a realistic
implementation featuring an SM-like Higgs boson, this model predominantely
leads to the existence of light additional Higgs bosons with a mass lower than
the SM-like Higgs boson.
