<p>Non-Hermitian (NH) systems can display exceptional topological defects without Hermitian counterparts, exemplified by exceptional rings in NH two-dimensional systems. However, exceptional topological features associated with higher-dimensional topological defects have only recently come into attention. We here investigate the topology of the singularities in an NH three-dimensional system. We find that the third-order singularities in the parameter space form an exceptional surface (ES), on which all three eigenstates and eigenenergies coalesce. Such an ES corresponds to a two-dimensional extension of a point-like synthetic tensor monopole. We quantify its topology with the Dixmier-Douady invariant, which measures the quantized flux associated with the synthetic tensor field. We further propose an experimentally feasible scheme for engineering such an NH model. Our results pave the way for investigations of exceptional topology associated with topological defects with more than one dimension.</p>