Using optical microscopy, we have studied the phase behavior of mixtures of 12- to 22-bp-long nanoDNA oligomers. The mixtures are chosen such that only a fraction of the sample is composed of mutually complementary sequences, and hence the solutions are effectively mixtures of single-stranded and double-stranded (duplex) oligomers. When the concentrations are large enough, such mixtures phase-separate via the nucleation of duplex-rich liquid crystalline domains from an isotropic background rich in single strands. We find that the phase separation is approximately complete, thus corresponding to a spontaneous purification of duplexes from the single-strand oligos. We interpret this behavior as the combined result of the energy gain from the end-to-end stacking of duplexes and of depletion-type attractive interactions favoring the segregation of the more rigid duplexes from the flexible single strands. This form of spontaneous partitioning of complementary nDNA offers a route to purification of short duplex oligomers and, if in the presence of ligation, could provide a mode of positive feedback for the preferential synthesis of longer complementary oligomers, a mechanism of possible relevance in prebiotic environments.