This paper presents the search for the production of a Higgs boson in association with a single top quark (tHq), using data collected in proton-proton collisions at a center-of-mass energy of 8 TeV corresponding to an integrated luminosity of 19.7 fb$^{−1}$. The search exploits a variety of Higgs boson decay modes resulting in final states with photons, bottom quarks, and multiple charged leptons, including tau leptons, and employs a variety of multivariate techniques to maximize sensitivity to the signal. The analysis is optimized for the opposite sign of the Yukawa coupling to that in the standard model, corresponding to a large enhancement of the signal cross section. In the absence of an excess of candidate signal events over the background predictions, 95% confidence level observed (expected) upper limits on anomalous tHq production are set, ranging between 600 (450) fb and 1000 (700) fb depending on the assumed diphoton branching fraction of the Higgs boson. This is the first time that results on anomalous tHq production have been reported.