Gemara

(Mishnah): If Reuven vowed not to benefit his son-in-law, he can give money to his daughter and say 'this is a gift to you, on condition that your husband has no Reshus over it. It is given only for you to buy food with it and to eat the food.'

(Rav): This works only when he says 'it is given only for you to buy food with it and to eat the food.' If he says 'do like you want with it', her husband acquires the money.

Question (R. Zeira): Here, Rav holds like R. Meir, who says that a husband acquires everything his wife gets;

Contradiction (a Stam Mishnah, which is assumed to be like R. Meir): To make Shituf Mavo'os (an Eruv to permit carrying in an alleyway on Shabbos), one takes a barrel of wine, and says that it is for everyone whose courtyard opens into the Mavoy.

Someone else is Mezakeh for them (acquires on their behalf). It can be the male or female (Yisrael) slave of the donor, or his adult son or daughter, or his wife.

(Summation of question): If a husband acquires all that his wife acquires, how can she be Mezakeh for others? The wine never left the husband's Reshus!

Answer #1 (Rava): R. Meir holds that a husband acquires everything his wife gets (from others). Regarding Shituf, since she acquires for them from her husband, she can acquire for others.

Objection (Ravina - Beraisa): One can be Mezakeh to others through his adult children, or his Hebrew slaves. He cannot be Mezakeh through his minor children, his Kena'ani slaves, or through his wife.

Answer #2 (Rav Ashi): The Mishnah (about Shituf) discusses a woman who owns a courtyard that opens into the Mavoy. Since she can acquire the wine for herself, she can acquire it also for the others.

Rishonim

The Rif and Rosh (Eruvin 25a and 7:8) bring the Mishnah.

Rosh: The Gemara in Nedarim says that he can be Mezakeh through his wife only when she has a house in the Chatzer. Migo (since) she can acquire for herself, she can acquire it also for the others. If not, she cannot.

Question #1: If he feeds his wife, R. Yehudah ben Beseira (Eruvin 73a) permits her without an Eruv (so there is no Migo). If he does not feed her, she can be Mezakeh for others even without a house in the Chatzer, for her hand is not like his regarding Metzi'os or earnings!

Answer #1 (Rosh): The case is, he gives to her money for food. Alternatively, he told her to feed herself from her earnings, and this suffices for her. Regarding Eruv, this is not considered feeding her (so she needs an Eruv). Nevertheless, her Yad is like his.

Question #2: If his slave has his own house in the Chatzer, R. Yehudah ben Beseira holds that they (need an Eruv, for otherwise they) forbid. If so, we should say that since the slave can acquire for himself, he can acquire also for others!

Answer (and Answer #2 to Question #1 - Rosh): The conclusion in Nedarim is like R. Yehudah ben Bava (in Eruvin), who permits the slaves and forbids the women when both of them are fed. However, above we ruled like R. Yehudah ben Beseira.

Rambam (Hilchos Eruvin 1:20): One can be Mezakeh food for an Eruv through his wife.

Poskim

Shulchan Aruch (OC 366:10): One can be Mezakeh food for an Eruv through his wife, even if he feeds her, and even if she does not have a house in the Chatzer.

Source (Gra DH v'Al and DH v'Yesh): The Gemara said 'Rav holds like R. Meir'. The Halachah follows Chachamim. He can be Mezakeh through her even when he feeds her. When he does not feed her, even R. Meir agrees! The opposing opinion (below) is of Tosfos, who rules like R. Meir.

Magen Avraham (17): Since someone else can give to her a gift on condition that her husband does not receive it, he can be Mezakeh through her.

Shulchan Aruch (ibid.): Some say that he cannot be Mezakeh through his wife if he feeds her or told her to feed herself from her earnings.

Beis Yosef (DH u'Mah she'Chosav v'Lo): Whether he gives to her food or money for food or told her to feed herself from her earnings, he gets her Metzi'os, so her Yad is like his.

Mishnah Berurah (65): This is only if her earnings suffice to feed her, for then he gets her Metzi'os.

Shulchan Aruch (ibid.): This is even if she has a house in the Chatzer.

Mishnah Berurah (66): She eats the food she receives in the house. Even so, she does not need an Eruv, therefore she cannot be Mezakeh for others.

Shulchan Aruch (ibid.): However, he can be Mezakeh through his wife if he does not feed her, even if she does not have a house in the Chatzer.

Gra (DH v'Afilu Yesh): We rule like the leniencies of R. Yehudah ben Beseira and of R. Yehudah ben Bava.

Shulchan Aruch (ibid.): It is good to be concerned for both opinions when possible.

Rema: B'Di'eved, we rely on the lenient opinion in Eruvin.