As sequence databases grow, characterizing diversity across extremely large collections of genomes requires the development of efficient methods that avoid costly all-vs-all comparisons. In addition to exponential increases in the amount of natural genomes being sequenced, improved techniques for the creation of human engineered sequences is ushering in a new wave of synthetic genome sequence databases that grow alongside naturally occurring genome databases. In this paper, we analyze the full diversity of available sequenced natural and synthetic plasmid genome sequences. This diversity can be represented by a data structure that captures all presently available nucleotide sequences, known as a pan-genome. In our case, we construct a single linear pan-genome nucleotide sequence that captures this diversity. To process such a large number of sequences, we introduce the plaster algorithmic pipeline. Using plaster we are able to construct the full synthetic plasmid pan-genome from 51,047 synthetic plasmid sequences as well as a natural pan-genome from 6,642 natural plasmid sequences. We demonstrate the efficacy of plaster by comparing its speed against another pan-genome construction method as well as demonstrating that nearly all plasmids align well to their corresponding pan-genome. Finally, we explore the use of pan-genome sequence alignment to distinguish between naturally occurring and synthetic plasmids. We believe this approach will lead to new techniques for rapid characterization of engineered plasmids. Applications for this work include detection of genome editing, tracking an unknown plasmid back to its lab of origin, and identifying naturally occurring sequences that may be of use to the synthetic biology community. The source code for fully reconstructing the natural and synthetic plasmid pan-genomes as well for plaster are publicly available and can be downloaded at https://gitlab.com/qiwangrice/plaster.git.