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There are several studies of highly-referenced researchers where a paired post-hoc test (e.g. Wilcoxon signed-rank) follows the omnibus unpaired test (e.g. Kruskal-Wallis):

From Huang et. al "THE BACH DOODLE: APPROACHABLE MUSIC COMPOSITION WITH MACHINE LEARNING AT SCALE"

A Kruskal-Wallis H test for both the number of P5s and P8s showed that there is at least one statistically significant difference between the four categories... Further, Mann-Whitney rank tests between the categories showed significant differences, ...

From Roberts et al. "A Hierarchical Latent Vector Model for Learning Long-Term Structure in Music"

... a Kruskal-Wallis H test of the ratings showed that there was a statistically significant difference between the models... A post-hoc analysis using the Wilcoxon signed- rank test with Bonferroni correction showed that participants rated samples...

A possible clue (which is not enough for me) arises from the last reference: Huang et al's "Music Transformer: generating music with long-term structure", where after a Kruskal-Wallis test, there are 2 post-hoc analysis, one with (paired) Wilcoxon signed-rank and another with (unpaired) Mann-Whitney U test.

A Kruskal-Wallis H test of the ratings showed that there was a statistically significant difference between the models... Table 5 show a post-hoc analysis on the comparisons within each pair, using the Wilcoxon signed-rank test for matched samples. Table 6 shows a post-hoc analysis of how well each model performed when compared to all pairs, and compares each model’s aggregate against each other, using the Mann–Whitney U test for independent samples.

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