In many, if not all, of the Nusach Sefard synagogues I've prayed at, the synagogue does not say Tachanun during Mincha during regular weekdays.
What is the source for this practice?
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See here where they suggest:
1 they often daven until after shkiya, and according to many opinions tachanun may not be said after shkiya, so a blanket rule was instituted so as never to come to saying after shkiya, which in some kabbalistic sources danger is associated with this practice.
2 tachanun by mincha requires intense concentration, which most people don’t have in the middle of their day
3 after it is night in Israel tachanun should not be said even in other parts of the world
Sources: Sefer Nimukei Orach Chaim, Shu”T Dvar Yehoshua Y:D 3:74, Zivchei Zedek siman 9 [minhag bagdad], Sefer Shulchan Hatohar Siman 22.