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Reassortment is an evolutionary process common in viruses with segmented genomes. These viruses can swap whole genomic segments during cellular co-infection, giving rise to novel progeny, i.e., the reassortant, formed from the mixture of parental segments. The reassortant can lead to severe illness, vaccine-elicited immunity escape, increased fitness, and access to new hosts, which pressures public health surveillance, disease control, and prevention. In light of the Markov Genealogy Process, we can simultaneously model the interdependent viral evolution of each segment tree in the context of a given epidemiological process, incorporated with genealogical changes associated with specified reassortment rates. In doing so, we argue that, in general, the number and pattern of reassortment events are not identifiable from segment trees alone, even with theoretically ideal data. We call this fact the fundamental problem of reassortment, which we illustrate using the concept of the "first-infection tree", a typically but not always counterfactual genealogy that would have been observed in the segment trees had no reassortment occurred. Further, we illustrate four problems that can arise logically in inferring reassortment events. Using simulated data, we show that these problems are common and can distort our perception of reassortment, even in small data sets. Finally, we discuss how existing methods can be augmented or adapted to account for the fundamental problem of reassortment and the four additional situations that can complicate the inference of reassortment.