How are DNA virus cladograms actually calculated in practice?
A cladogram, or phylogenetic tree1, is constructed by comparing similarities and differences between organisms, and placing those within an evolutionary model.2 In practice, especially for viruses and microorganisms, the analysis is performed on the organisms' nucleic acid or protein sequences.3,4 For example, a simple model might assume that all nucleic acid changes are equally likely; others might treat transitions and transversions differently; still others may have a unique substitution rate for each possibility, with the overall likelihood depending on the gene under analysis, position within the gene, frequency of each nucleotide, etc. The choice of model may be directly selected by the researcher, but is often selected from an analysis that balances model fit against over-parameterization. The researcher selects the tree that optimizes some aspect of the model, for example the likelihood that the selected tree could produce the observed data under the chosen evolutionary model.5,6
Is the procedure different for RNA viruses?
No, the procedure outlined above applies to both DNA and RNA viruses, although aspects of the model parameters may differ.
Are these processes somewhat subjective?
The same model, applied to the same data, will produce the same result (assuming a thorough tree search was performed), so that aspect is not subjective. However, model choice implies assumptions about the nature of biological evolution, and researchers may disagree on the validity of those assumptions.
That being said, the methods for analyzing viral outbreaks are very well characterized and it's unlikely two competent researchers will produce greatly conflicting trees. In reference to the monkeypox example, the two clades mentioned are strongly supported, to the point where it would be innaccurate to call them subjective.
It should be noted that for any given tree, the identification of clades is not at all subjective. Every node in a phylogeny represents an ancestor of a clade. However, researchers find some clades especially relevant to the study system, so they will call them out with a name or label. For example in monkeypox the West African/Central African clades are very relevant to most discussions.
The above generally applies to any group of organisms. Viruses and other microbes may differ in that we have access to specimens that were the actual ancestors of the current generation. For example, apart from what we learn from fossils, we can only infer the characteristics of the common ancestor of chimps and humans. However, for SARS-CoV-2 we have samples of the original isolate, even if viruses with that exact sequence are no longer present in the wild. When constructing a tree with this sample we would place that specimen at the node representing the ancestor of all current SARS-CoV-2, rather than at one of the tips of the tree.
Finally the phrase "basal clade" is nonsensical. All clades, by definition, contain an ancestor and all of its descendants. There are clades where the ancestor existed earlier in time and has more descendants than others, but then, that clade is itself just a part of a larger clade. The linked article states, "a basal clade is one closest to the root common ancestor", but of course, that ancestor had more than one successful descendant, and so both descendants are equally close to the root. The article also states, "The S clade... mutated to produce the V clade," but if the term is being used properly the V clade is a part of, and a subclade within, the S clade.7 Here is a longer discussion of why "basal" is usually an inappropriate descriptor of phylogenetic placement. (However, as mentioned above, for some microbes we do actually have samples of the ancestral strains, so "basal is a bit more meaningful there.)
1A phylogenetic tree is a broader category of analysis where the lengths of the branches in the tree have meaning. A cladogram only shows topological relationships, while a phylogenetic tree shows both topology and some magnitude of change (usually either evolutionary or temporal).
2A tree can also be produced by simple similarity clustering, which is fast, but can be positively misled compared to the true phylogeny.
3Phylogenies can also be constructed from other information, such as morphological similarities/differences, as in fossils, for example. Still, some evolutionary model is used to transform the similarity matrix into the phylogeny.
4Of course, the protein sequence can be derived from the nucleic acid sequence. Which is used usually depends on the nature of the question and the breadth of the organisms included in the analysis.
5This is the maximum likelihood method, Bayesian methods are also common.
6Selection of the optimal tree is NP-hard, and there exist many methods to tractably estimate the solution.
7SARS-CoV-2 has its clades named after specific mutations that "define" (are synapomorphies for) the clade. However, this isn't strictly informative, since there's no reason a member of that clade couldn't mutate back to the previous state, or that a member of another clade couldn't pickup an identical mutation.