A plausible process for non-enzymatic RNA replication would greatly simplify models of the transition from prebiotic chemistry to simple biology. However, all known conditions for the chemical copying of an RNA template result in the synthesis of a complementary strand that contains a mixture of 2′-5′ and 3′-5′ linkages, rather than the selective synthesis of only 3′-5′ linkages as found in contemporary RNA. Here we show that such backbone heterogeneity is compatible with RNA folding into defined three-dimensional structures that retain molecular recognition and catalytic properties and, therefore, would not prevent the evolution of functional RNAs such as ribozymes. Moreover, the same backbone heterogeneity lowers the melting temperature of RNA duplexes that would otherwise be too stable for thermal strand separation. By allowing copied strands to dissociate, this heterogeneity may have been one of the essential features that allowed RNA to emerge as the first biopolymer.