Tissue culture regenerants display frequent unexpected changes resulting in plants with novel characteristics, called somaclonal variants. At lower frequency, similar variants called sports arise during normal growth. These variants can have great economic value, but relatively little is known of their causes, nature and evolutionary significance. This knowledge gap is even greater in polyploids, where genetic redundancy and buffering of gene dosage imbalance permits higher variation. To investigate somatic genomic remodeling in a polyploid, we compared protoplast-regenerants of tetraploid potato cultivar Desiree and its natural clonal variants. We document single-nucleotide and structural variation among natural isolates of Desiree. In protoplast-regenerated clones, we found chromosome structural variation consistent with homologous and non-homologous DNA repair that resulted in copy-neutral loss of heterozygosity. Interestingly, we also found evidence that the chimeric state of the parental clone contributed to variation. Our observations provide a framework for understanding genome instability and its consequences in potato and other vegetatively propagated polyploids.