A common approach for parameterising eddy transport of passive tracers by mesoscale eddies in the ocean is by invoking a transport tensor. The symmetric part of this tensor, the diffusion tensor, quantifies diffusive eddy tracer transport. Recent studies have diagnosed opposite-signed eigenvalues (diffusivities) of the diffusion tensor from eddy-resolving simulations, while all current parameterisations implement only positive diffusivities. For opposite-signed eigenvalues the associated diffusive eddy tracer flux is not necessarily down-gradient and therefore may not mix the tracer by transferring variance to the small scales. In this study we explore such diffusive eddy fluxes by using an eddy-resolving simulation of passive tracers with a relaxation (source/sink) forcing. After confirming that the diffusion tensors for different tracer pairs have opposite-signed eigenvalues, we show that the corresponding diffusive eddy tracer flux drives a net down-gradient transfer of variance, as would be guaranteed when the diffusion tensor eigenvalues are both positive. Locally up-gradient fluxes are common, with their frequency strongly dependent on the relaxation profile. The effects of weakening/strengthening the relaxation on the frequency of down-gradient fluxes is different for each tracer. However, for all tracers considered the amplitude of the net down-gradient transfer weakens as the relaxation strengthens, a consequence of the homogeneous diffusion dissipating less eddy variance. Our results indicate that for oceanic tracers with sources/sinks the parameterised diffusive eddy tracer fluxes should not be globally down-gradient.