  We study a version of Two Higgs Doublet Models with non-standard flavor
violation in the up quark sector. We find branching ratios for the rare top
decays $t\to hc$ and $t\to hu$ that are within reach of current and future
colliders, while other flavor constraints from rare $B$ decays and neutral $D$
meson mixing, as well as constraints from Higgs signal strength measurements
remain under control. The most prominent collider signature of the considered
setup is $pp \to t H \to tt \bar c$, providing continued motivation to search
for same-sign tops at the LHC as well as a simple framework to interpret these
searches. As a byproduct of our study, we provide updated SM predictions for
the rare top decays BR$(t \to h c)_\text{SM} = (4.19^{+1.08}_{-0.80} \pm 0.16)
\times 10^{-15}$ and BR$(t \to h u)_\text{SM} = (3.66^{+0.94}_{-0.70} \pm 0.67)
\times 10^{-17}$ with the main uncertainties coming from higher order QCD and
CKM matrix elements.
