Gemara

Gitin 47a (Mishnah): If one sold his field to Nochrim, for Tikun ha'Olam he buys the fruit and brings from it Bikurim (the first Peros to ripen of the seven species. They are given to a Kohen in the Mikdash.)

(Rabah): Ein Kinyan l'Nochri b'Eretz Yisrael (a Nochri cannot acquire land in Eretz Yisrael to exempt what grows from Ma'aser). "To Me (Hash-m) is the land."

(R. Elazar): Yesh Kinyan l'Nochri b'Eretz Yisrael - "(you will tithe) your grain", not the grain of Nochrim.

R. Elazar expounds "your Degen (grain)", not grain of Nochrim. Rabah expounds "your Digun (Miru'ach, final processing of grain)", not the Digun of Nochrim.

Menachos 31a (Beraisa - R. Shimon Shezuri) Version #1: Once, Tevel became mixed with Chulin. R. Tarfon told me to buy (Demai, i.e. Peros of an Am ha'Aretz), and to separate Ma'aser from it for the mixture. (The Tevel was Batel in a majority of Chulin, so it must be tithed only mid'Rabanan, like Demai.)

R. Tarfon didn't tell him to buy from a Nochri, for he holds that Ein Kinyan, and a Nochri's produce must be tithed mid'Oraisa. One may not separate from what is liable to exempt produce that is exempt (mid'Oraisa).

Version #2: R. Tarfon told him to buy Peros Nochri, and to separate from it on the mixture. He holds that Yesh Kinyan, so it is exempt mid'Oraisa.

(Rav Papa): The Halachah follows R. Shimon Shezuri.

66b (Rava): Tana'im argue about Miru'ach of a Nochri:

(Beraisa - R. Meir and R. Yehudah): We may take Terumah from anyone's Peros to exempt anyone else's Peros (it is all Chayav mid'Oraisa);

R. Yosi and R. Shimon say, we may not take Terumah from a Yisrael's Peros on a Nochri's (for they are exempt mid'Oraisa), or vice-versa.

67a: Miru'ach of a Nochri is obligated mid'Rabanan. This is a decree due to rich people (lest they buy from a Nochri merchant who did Miru'ach, and people will think that also what a Yisrael merchant sells is exempt. Alternatively, perhaps rich people will give their Peros to Nochrim for Miru'ach, to be exempt from Ma'aser.)

Bechoros 11a (R. Shmuel bar Noson): If one buys Tevel from a Nochri after Miru'ach, he separates Ma'aseros and keeps them.

Question: If the Nochri did Miru'ach, it is exempt from Ma'aser! "Diguncha" excludes Miru'ach Nochri.

Answer: The Yisrael did Miru'ach while the produce belonged to the Nochri. He must Ma'aser them, for Ein Kinyan l'Nochri. He keeps the tithes, for a Kohen could not have forced the Nochri to give. (They are due to the Yisrael, who bought from the Nochri.)

Rishonim

Rambam (Hilchos Terumos 1:11): If a Nochri bought land in Eretz Yisrael, and he owned the Peros at the time of final processing and he did Miru'ach, they are totally exempt. It says '"your Digun", not the Digun of Nochrim. If a Yisrael bought them after they were detached and before final processing, and he finished them, they are obligated mid'Oraisa. He gives Terumah Gedolah to a Kohen. He sells Terumas Ma'aser to a Kohen, and keeps the Ma'aser Rishon. Regarding Ma'aser Rishon and Terumas Ma'aser, he can tell the Levi and Kohen 'I received the Peros from someone from whom you could not have taken anything.'

Rambam (Hilchos Ma'aser 11:15): If a Yisrael deposited finished Chulin with a Nochri, he need not separate anything when he gets it back. Even if the Nochri switched it with his own produce (, it is exempt, like we said in Hilchos Terumos.

Ra'avad: It is obligated mid'Rabanan.

Kesef Mishneh: The Rambam learns from Menachos 67a that Miru'ach Nochri exempts even mid'Rabanan. Chachamim decreed to obligate it due to rich people. Also, the question in Bechoros 11a shows that it is totally exempt. R. Shimon (Demai 3:4) obligates one who deposited Chulin with a Nochri to tithe what he gets it back like Demai. He agrees that Miru'ach Nochri totally exempts. We are concerned lest he switched the Peros with those of another Yisrael.

Gra (YD 331:9): Tosfos holds that even though a Mishnah (Demai 5:9) obligates Miru'ach Nochri, this is not the Halachah, for that the one who holds that Ein Kinyan exempts Miru'ach Nochri (Gitin 47a). The Rambam follows the simple reading of the Gemara. The Gemara obligates Miru'ach Nochri mid'Rabanan. The Rambam explains that this is when it grew in a Yisrael's field.

Poskim

Shulchan Aruch (YD 331:4): If a Nochri owned Peros of his land at the time of final processing and he did Miru'ach, they are totally exempt. If a Yisrael bought them after they were detached and before final processing, and he finished them, they are obligated. He separates Terumos and Ma'aser Rishon. He keeps the Ma'aser Rishon, for the Levi could not have forced the Nochri to give to him.

Beis Yosef (DH u'Mah she'Chosav Al): Tosfos (Bava Metzi'a 101a DH Savruha) learns from Menachos 31a that according to the opinion that Yesh Kinyan, if a Yisrael bought the produce, it is obligated mid'Rabanan. The Ra'avad agrees. The Rambam exempts even mid'Rabanan, even though he holds that Ein Kinyan. Also Semag says in the name of R. Shimshon that perhaps it is exempt even mid'Rabanan nowadays, since even a Yisrael's land is not obligated mid'Oraisa, according to R. Yosi. However, the Rambam exempts only when a Nochri owned them during final processing. If a Yisrael bought them after they were detached and before final processing, and he finished them, they are obligated mid'Oraisa.

Rema: He sells the Terumas Ma'aser to a Kohen and keeps the money.

Beis Yosef (4 DH Leshon): The Rambam learns from Bechoros 11a.

Shulchan Aruch (7): If a Yisrael sold his Peros to a Nochri after Onas ha'Ma'aseros, even though the Nochri finished them, they are totally obligated.

Gra (16): The Shulchan Aruch did not discuss whether or not Miru'ach of a Nochri exempts Peros that a Yisrael grew, for in any case it is obligated mid'Rabanan, and nowadays all Terumos and Ma'aseros are mid'Rabanan.