DNA in solution can be condensed into dense aggregates by multivalent counterions. Here we investigate the effect of a nearby surface on the morphology of DNA condensates. We show that, contrary to what has often been assumed, interactions between DNA condensates and the surface can strongly influence the observed morphology. This limits the usefulness of surface probes such as atomic force microscopy for studying the morphology of condensates in bulk solution. Surprisingly, we find that the most negatively charged surface disturbs the condensate morphology most, suggesting that the microscopic mechanism resulting in DNA condensation is also responsible for the attractive force between DNA and the surface.