Based on something that really happened last week:
After completing Mincha, the Shliach Tzibur realized he was bareheaded the entire time. Nobody in the congregation had noticed it.
Has the congregation fulfilled its duty or should somebody repeat the Kadeishim and Chazarat HaShatz again?
Looking for sources or solid proofs.
I can't seem to find anything specific - here's related Halachot:
The Kitzur Shulchan Aruch in 3:6 states that it's forbidden to walk 4 cubits or to say anything Holy while bareheaded.
אָסוּר לֵילֵךְ אַרְבַּע אַמּוֹת אוֹ לְהוֹצִיא מִפִּיו דָּבָר שֶׁבִּקְדֻשָּׁה בְגִלּוּי הָראֹשׁ.
But he doesn't say what the state of the Holy sayings are if they were said bareheaded.
The Shulchan Aruch in 91:3 says that some say that one shouldn't mention Gcd's name while bareheaded, and some say one should protest if someone enters a synagogue bareheaded.
ג: יֵשׁ אוֹמְרִים שֶׁאָסוּר לְהוֹצִיא הַזְכָּרָה מִפִּיו בְּרֹאשׁ מְגֻלֶּה, וְיֵשׁ אוֹמְרִים שֶׁיֵּשׁ לִמְחוֹת שֶׁלֹּא לִכָּנֵס בְּבֵית הַכְּנֶסֶת בְּגִלּוּי הָרֹאשׁ.
This makes it sound like it's not imperative. But in 2:6 he says it's forbidden to walk 4 cubits while bareheaded.
ו אָסוּר לֵילֵךְ בְּקוֹמָה זְקוּפָה וְלֹא יֵלֵךְ ד' אַמּוֹת בְּגִלּוּי הָרֹאשׁ:
It seems like there is an opinion that one can sit (or stand still), remove one's head covering and make then a Bracha. In that case, the congregation would have nothing to worry about, ex post facto.