CeRhIn$_5$ provides a textbook example of quantum criticality in a heavy fermion system: Pressure suppresses local-moment antiferromagnetic (AFM) order and induces superconductivity in a dome around the associated quantum critical point (QCP) near $p_{c} approx 23,$kbar. Strong magnetic fields also suppress the AFM order at a field-induced QCP at $B_{rm c}approx 50,$T. In its vicinity, a nematic phase at $B^*approx 28,$T characterized by a large in-plane resistivity anisotropy emerges. Here, we directly investigate the interrelation between these phenomena via magnetoresistivity measurements under high pressure. As pressure increases, the nematic transition shifts to higher fields, until it vanishes just below $p_{rm c}$. While pressure suppresses magnetic order in zero field as $p_{rm c}$ is approached, we find magnetism to strengthen under strong magnetic fields due to suppression of the Kondo effect. We reveal a strongly non-mean-field-like phase diagram, much richer than the common local-moment description of CeRhIn$_5$ would suggest.
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