It is conceivable that an RNA virus could use a polysome, that is, a string of ribosomes covering the RNA strand, to protect the genetic material from degradation inside a host cell. This paper discusses how such a virus might operate, and how its presence might be detected by ribosome profiling. There are two possible forms for such apolysomally protected virus, depending upon whether just the forward strand or both the forward and complementary strands can be encased by ribosomes (these will be termed type 1 and type 2, respectively). It is argued that in the type 2 case the viral RNA would evolve anambigrammaticproperty, whereby the viral genes are free of stop codons in a reverse reading frame (with forward and reverse codons aligned). Recent observations of ribosome profiles of ambigrammatic narnavirus sequences are consistent with our predictions for the type 2 case. Creative Commons Attribution license.