The chazan skipped Tachanun, and everyone assumed there was a chatan or a brit. After davening, the chazan said he just forgot Tachanun. People disagreed about whether we could/should say Tachanun at that point. What is the halacha? Answer:
The Shulchan Aruch (Orach Chayim 131:1) says that one must not speak between Shemoneh Esrei and Tachanun, based on "students of the Rashba’s" (see Beit Yosef, OC 131) comment on the following gemara (Bava Metzia 59b). After Rabbi Eliezer’s major dispute with his brother-in-law, Rabban Gamliel, the former’s wife was afraid that the intensity of his Tachanun could cause harm to her brother, so she always interrupted him when it was time for Tachanun. The Rashba reasons that she could not have prevented him from saying Tachanun all day, but just made him stop and/or speak at the right time, to lower its efficacy. This taught the Shulchan Aruch and others of the danger of interruptions at that time.
What does the above teach us about the required level of connection between Shemoneh Esrei and Tachanun? The conviction that a break makes Tachanun less effective does not necessarily mean that Tachanun need not or should not be said after such a break or that it lacks value. The Rashba/Shulchan Aruch’s understanding of the story of Rabbi Eliezer strongly implies that R. Eliezer recited Tachanun after the break. The Rivash (412) claimed that his wife bothered him until he forgot to say it, also implying he would have said it later. Thus, at this point, we would say: "Better late than never."
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The Taz (OC 131:10, which seems to contradict Taz, Yoreh Deah 376:2) complicates the matter. He discusses whether non-mourners who daven at an avel’s house, where Tachanun is omitted because the presence of "strict judgment" makes it not worthwhile to recite Tachanun there, should make it up when they get home. He says not to do so based on the halacha that Tachanun should come without an interruption after Shemoneh Esrei. It is unclear if that means it is not required or wrong (there are kabbalistic sources for such a possibility – see Shulchan Hatahor 131:16), unnecessary, or somewhere in between. This seemingly indicates that you would not say Tachanun, in your case, at the end of tefilla. (Change of place does not seem to be the issue – see Magen Avraham 131:1).
However, the Taz’s claim is surprising, considering the indications from the gemara and the p’sak (Mishna Berura 131:2) that b’di’eved, if one made a break, he says Tachanun anyway. How could the gemara’s case be a model for a ruling not to say Tachanun at all? The L’horot Natan (VI:7) raises the possibility that continuing tefilla is worse than talking, and in the Taz’s case (and ours), it could be too late for Tachanun, not just of reduced value. However, he posits that this is not so and that the Taz would agree in our case to say Tachanun. Here, at the time of Tachanun, there was an obligation to recite it, which was pushed off on technical grounds (the chazan’s mistake). The Taz spoke only about a case that at the correct time, there was no obligation (albeit based on the circumstances). What he says is that it is not created later at an unnatural time (which, in turn, we learn from the halacha that it is important not to break).
The Derech Hachayim (42:(7)) implies that the Taz would not say Tachanun after any break. However, the Derech Hachayim (42:1) and Eliya Rabba (OC 131:1), who are accepted by the Mishna Berura (131:2), reject this view. Rav SZ Auerbach is also cited (Halichot Shlomo 11:2) as instructing to say Tachanun if it was accidentally skipped, even after laining, and presumably also after davening.
Some contemporary poskim (Ishei Yisrael 26:(1); Dirshu 131:3) cite an account about the Chazon Ish and a very cryptic reaction of Rav Chaim Kaniefsky which may indicate to not say Tachanun once Chatzi Kaddish was said. While the stakes are low (see Rivash ibid.) in both directions, we recommend saying Tachanun if it was skipped by mistake, as this approach has a stronger basis in the sources/logic.