Coronavirus mRNA can recombine within a cell. Their transcription is already error prone, and recombination adds extra level of mixing and matching to generate variants and evolve. But normally this occurs with only the one variant that infects a subject and it's part of how variants arise, whether they are advantageous for the virus or not.
But in theory -
in cells in vitro and in
at least one documented patient in France - if two variants infect at the same time and enter the same cell (which is unlikely but increases in likelihood the more people are infected), recombination could occur across two different variant mRNAs.
From what can gather, this should be rare. The infection would really have to be almost simultaneous because the response for one should fight the other upon entry. And infect the exact same cell. Two variants in the body infecting different cells (which is mathematically most likely if co-infection does happen) wouldn't lead to recombination across variants. Maybe immunosuppressed individuals could be a vector where this could be more likely?
Note that were a person infected by two different variants at the same time, they will not necessarily be more sick. it really depends on their immune system (vaccination status + prior exposure + general health of the immune system) vs. how big of a viral load they were exposed to.