Dear all,

After me and my colleagues worked on the first paper you mention, I recently created another merge-based sorting algorithm, which I called "Adaptive Shivers Sort". This is a close variant of the Augmented Shivers Sort presented by Buss & Knop in their article
https://arxiv.org/abs/1801.04641

Like Munro & Wild's Powersort and Peeksort, this algorithm has an optimal worst-case running time (up to a linear additive constant) when measured with the merge cost model that is used in all the articles you cite in this discussion. It also maintains a stack of size log_2(n).
I could not prove such an optimality result for Buss & Knop Augmented Shivers Sort.

Custom tests that I had performed suggest that this algorithm is, in practice, as efficient as Munro & Wild's algorithms, and also does not suffer from critically bad cases.

Moreover, like Buss & Knop's 2-merge, and unlike Munro & Wild's Powersort and Peeksort, this algorithm has the advantage of having a structure that is very similar to that of Timsort, which may be an advantage in your situation.

That is why, upon reading your discussion, I refurbished my notes about Adaptive Shivers Sort, which you may find here:
http://igm.univ-mlv.fr/~juge/papers/shivers_arxiv.pdf

These notes include a description of the algorithm and a worst-time complexity analysis. If extending my analysis of this algorithm or investigating tuning details were of interest for you, I would be happy to help.

Best regards,

Vincent Jugé