Positive-sense single-stranded RNA viruses form the largest and most diverse group of eukaryote-infecting viruses. Their genomes comprise one or more segments of coding-sense RNA that function directly as messenger RNAs upon release into the cytoplasm of infected cells. Positive-sense RNA viruses are generally accepted to encode proteins solely on the positive strand. However, we previously identified a surprisingly long (∼1,000-codon) open reading frame (ORF) on the negative strand of some members of the family Narnaviridae which, together with RNA bacteriophages of the family Leviviridae, form a sister group to all other positive-sense RNA viruses. Here, we completed the genomes of three mosquito-associated narnaviruses, all of which have the long reverse-frame ORF. We systematically identified narnaviral sequences in public data sets from a wide range of sources, including arthropod, fungal, and plant transcriptomic data sets. Long reverse-frame ORFs are widespread in one clade of narnaviruses, where they frequently occupy >95 per cent of the genome. The reverse-frame ORFs correspond to a specific avoidance of CUA, UUA, and UCA codons (i.e. stop codon reverse complements) in the forward-frame RNA-dependent RNA polymerase ORF. However, absence of these codons cannot be explained by other factors such as inability to decode these codons or GC3 bias. Together with other analyses, we provide the strongest evidence yet of coding capacity on the negative strand of a positive-sense RNA virus. As these ORFs comprise some of the longest known overlapping genes, their study may be of broad relevance to understanding overlapping gene evolution and de novo origin of genes.