אורח חיים ס״וOrach Chayim 66

Se’if 1א׳

בֵּין הַפְּרָקִים שׁוֹאֵל בִּשְׁלוֹם אָדָם נִכְבָּד וּמֵשִׁיב שָׁלוֹם לְכָל אָדָם, וּבָאֶמְצַע שׁוֹאֵל בִּשְׁלוֹם מִי שֶׁהוּא יָרֵא מִמֶּנּוּ, כְּגוֹן אָבִיו אוֹ רַבּוֹ, אוֹ מִי שֶׁהוּא גָּדוֹל מִמֶּנּוּ בְּחָכְמָה, וְכָל שֶׁכֵּן מֶלֶךְ אוֹ אַנָּס. וּמֵשִׁיב שָׁלוֹם לְאָדָם נִכְבָּד, וַאֲפִלּוּ בְּאֶמְצַע הַפָּסוּק, חוּץ מִפָּסוּק שְׁמַע יִשְׂרָאֵל וּבָרוּךְ שֵׁם כְּבוֹד מַלְכוּתוֹ לְעוֹלָם וָעֶד, שֶׁלֹּא יַפְסִיק בָּהֶם כְּלָל, אִם לֹא מִפְּנֵי מִי שֶׁיָּרֵא שֶׁמָּא יַהַרְגֶנּוּ:

[The Laws Regarding] In Which Place One May Interrupt and in Which Place One May Not Interrupt. Containing 10 Se-ifim Between the sections [of Shema or it's blessings], one may greet an honored person and respond to a greeting from any person. And in the middle [of a section], one may greet someone that one reveres, such as one's father, one's teacher, or one greater than oneself in wisdom, and all the more so a king or one who compels. And one may respond to a greeting from an honored person, even in the middle of a verse, except for the verse of "Sh'ma Yisrael" and "Baruch shem kavod malchuto l'olam va-ed", during which one may not interrupt at all unless it is out of fear that he will be killed.

Se’if 2ב׳

אִם שָׁכַח לְהָנִיחַ צִיצִית וּתְפִלִּין יָכוֹל לְהַפְסִיק בֵּין הַפְּרָקִים לַהֲנִיחָם, וִיבָרֵךְ עֲלֵיהֶם: הַגָּה: וְיֵשׁ אוֹמְרִים שֶׁלֹּא יְבָרֵךְ עֲלֵיהֶם עַד אַחַר הַתְּפִלָּה, וְהָכֵי נָהוּג לְעִנְיַן טַלִּית (בֵּית יוֹסֵף).

If one forgot to don tzitzit or tefillin, one can interrupt between sections [of Shema or it's blessings] and don them, and one can make a blessing over them. Rem"a: And there are those who say one should not make a blessing over them until after the Prayer [Amidah], and such is the custom with regard to tallit (Beit Yosef)

Se’if 3ג׳

לְקַדִּישׁ וְלִקְדֻשָּׁה וּלְבָרְכוּ מַפְסִיק אֲפִלּוּ בְּאֶמְצַע הַפָּסוּק, וְכֵן לְמוֹדִים, אֲבָל לֹא יֹאמַר אֶלָּא תֵּבַת מוֹדִים בִּלְבַד: הַגָּה: וְכֵן בְּבָרְכוּ לֹא יֹאמַר יִתְבָּרַךְ וְיִשְׁתַּבַּח וְכו', וְיֵשׁ אוֹמְרִים דְּאָמֵן שֶׁעוֹנִין אַחַר בִּרְכַּת הָאֵל הַקָּדוֹשׁ וְאַחַר שׁוֹמֵעַ תְּפִלָּה יֵשׁ לוֹ דִּין קְדֻשָּׁה, וְיוּכַל לַעֲנוֹת אוֹתָם בִּקְרִיאַת שְׁמַע וְכֵן עִקָּר וּלְכָל ה"מ פּוֹסְקִים מִכָּל שֶׁכֵּן בְּאוֹמֵר תַּחֲנוּנִים (תְּשׁוּבַת הָרַשְׁבָּ"א).

For Kaddish, K'dushah, and Bar'chu, one interrupts [Shema or it's blessings] even in the middle of a verse. And so too for Modim, but one [should] only say the word "Modim" alone. Rem"a: And also, regarding [answering] Barchu [between or in the middle of sections] (which is allowed), one should not say "yitbarach v'yishtabach, etc." (ie. there is an old custom in some places where the congregation would say "yitbarach v'yishtabach, etc." while the leader is saying Barchu). And there are those who say that the "Amen" that we respond with after the blessing of "ha-El ha-Kadosh" ("the Holy God" - at the end of the K'dushah in Shemona Esrei) and after "shomeya tefilah" ("the one who hears prayer" in Shemona Esrei) has the status of the K'dushah, and one can answer them during the Recitation of the Sh'ma. And that is the main opinion [that should be followed]. And with all these things, we interrupt, and certainly when saying supplications [Teshuvot of the Rashba]