  We discuss the $Z_2$ symmetric two Higgs doublet model with a real soft
breaking term (real 2HDM). We explain in detail why it is not tenable to assume
CP conservation in the scalar sector to keep the dimension two term real, while
CP is violated by the dimension four Yukawa couplings. We propose the
calculation of the infinite tadpole of the (would-be) pseudoscalar neutral
scalar. We construct a simple toy model with the same flaws, where the
unrenormalizable infinity is easier to calculate. We then turn our attention to
the same tadpole in the real 2HDM. We spearhead this effort focusing on
diagrams involving solely bare quantities. This involves hundreds of Feynman
three-loop diagrams that could feed the CP violation from the quark into the
scalar sector, and is only possible with state of the art automatic computation
tools. Remarkably, some intermediate results agree when using three independent
derivations, including the peculiar cancellation of the leading pole divergence
due to a subtle interplay between masses and the Jarlskog invariant, which we
calculate analytically. The calculation is not complete however, since the full
two-loop renormalization of the real 2HDM is not yet available in the
literature. Still, we argue convincingly that there is an irremovable infinity.
