  Current Higgs data show an ambiguity in the value of the Yukawa couplings to
quarks and leptons. Not so much because of still large uncertainties in the
measurements but as the result of several almost degenerate minima in the
coupling profile likelihood function. To break these degeneracies, it is
important to identify and measure processes where the Higgs coupling to
fermions interferes with other coupling(s). The most prominent example, the
decay of $h \to \gamma \gamma$, is not sufficient to give a definitive answer.
In this Letter, we argue that $t$-channel single top production in association
with a Higgs boson, with $h\to b\bar b$, can provide the necessary information
to lift the remaining degeneracy in the top Yukawa. Within the Standard Model,
the total rate is highly reduced due to an almost perfect destructive
interference in the hard process, $W b \rightarrow t h$. We first show that for
non-standard couplings the cross section can be reliably computed without
worrying about corrections from physics beyond the cutoff scale $\Lambda\gtrsim
10\,\mathrm{TeV}$, and that it can be enhanced by more than one order of
magnitude compared to the SM. We then study the signal $ p p \rightarrow t h j
(b)$ with 3 and 4 $b$'s in the final state, and its main backgrounds at the
LHC. We find the 8 TeV run dataset to be sensitive to the sign of the anomalous
top Yukawa coupling, while already a moderate integrated luminosity at 14 TeV
should lift the degeneracy completely.
