Which berachah acharonah should one say first: borei nefashot, or me'en shalosh?
(Please cite your sources.)
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Biur Halacha 202:11 says in the name of the Pri Megadim to first say Me'en Shalosh and then Borai Nefashos since Me'en Shalosh is more Chashuv.
Despite all the discussion about the proper order of brachot before food, there is very little discussion about the order of brachot after food. One piece (the only one I can find), is in a gloss of Rabbeinu Peretz to the Semak (151:21) which is also quoted in the Agudah (Brachot 149) which recommends saying Borei Nefashot before Al haMichya in the case where you ate a non-seven-species fruit and had a mezonot food.
The Magen Avraham (OC 202:26) quotes this Agudah, but suggests that perhaps that order was specific to that case where one would be in doubt if the Al haMichya exempts the apple or not. (The doubt would relate to a different nusach of the blessing than what we use.) The Magen Avraham leaves unresolved (צ"ע) whether the Agudah's rule is meant to be general or not.
The Peri Megadim (EA 202:25-6) suggests two reasons why Me'ein Shalosh should precede Borei Nefashot:
It is a more specific bracha, and in the laws of brachot before food, specific brachot generally get precedence.
A minority opinion (quoted in Magen Avraham OC 172:3) holds that Me'ein Shalosh is a biblical requirement and should precede Borei Nefashot which is rabbinic.
The Aruch haShulchan (OC 202:40) rules that the Agudah's rule is not meant to be a general one (IMHO a very reasonable read of the Agudah), and accordingly rules that in general one may say a Me'ein Shalosh and Borei Nefashot in whatever order one pleases.
The Shulchan Aruch 208:13 rules that if one ate fruit from the 7 species plus apples, you don't need to say borei nefashot on the apples, because it's covered by the beracha of "al haetz", but if you had wine and apples (or meat or p'ri ha'adamah), you need to say both berachot.
Kaf HaHayyim 208:73 brings the opinions of the Semak and the Agudah who are concerned that the words "v'al t'nuvat hasadeh" in me'en shalosh cover other foods that grow from the ground (even if they aren't among the 7 species) and therefore l'chatchilah you should say borei nefashot first if all of the food that you're saying borei nefashot on is p'ri ha'adamah. If you said me'en shalosh first, you can still say borei nefashot afterwards, because the majority opinion isn't concerned about the words "v'al t'nuvat hasadeh."
Rav Ovadia and sons (Halacha Berurah 208:50 , Yalkut Yosef 208:6) also rule that on the p'ri ha'adamah, one should say borei nefashot first. Halachah Berurah disagrees with the Kaf HaHayyim about what to do if you already said me'ein shalosh first -- in that case, he holds that you cannot say borei nefashot afterward. (But Yalkut Yosef agrees with the Kaf HaHayyim.) They also say that if you had meat as well as food from the 7 species, you should say me'en shalosh first, and borei nefashot second.