You could use the 'Multiple Comparisons Between Treatments' procedure described in Siegel & Castellan , Nonparametric Statistics, 2nd Ed., 213-215.

See:

https://myaseen208.com/agricolae/reference/kruskal.html

or the Dunn test:

https://statsandr.com/blog/kruskal-wallis-test-nonparametric-version-anova/

See also the R Companion by Salvatore Magniafico:

https://rcompanion.org/handbook/F_08.html

Etc. just Google 'kruskal-wallis multiple comparisons'

Agreeing with Jos Feys . It really makes no sense to use Tukey HSD as a post-hoc to Kruskal-Wallis, because they have different assumptions and handle and analyze the data in very different ways. The Dunn (1964) test is the most common post-hoc test for Kruskal-Wallis. As mentioned in the rcompanion.org link, there are also Conover and Nemenyi tests for this purpose.

A better question might be why are you doing kruskal Wallace followed by something else? Why not just test your hypothesis and be done with it. I think I know a reference let me check David Booth

Here's some modern tests and they don't require that they follow anything. What is the purpose of Kruskal Wallace anyway.? Best wishes David Booth

Also see Jochen Wilhelm's comment on this question. See attached screenshot. Best wishes David Booth

Here's the rest of the comment that is relevant to your question. David Booth