Gemara

Rav and Shmuel were eating. Rav Simi bar Chiya came, and ate quickly.

Rav: Do you want to join for a Zimun? We already finished!

Shmuel: If they would bring (for dessert) mushrooms for me and young fowl for you, wouldn't we eat?! (Therefore, it is as if we are still eating, so Rav Simi may join for a Zimun.)

Chulin 86b: Rav Beruna and Rav Chananel were eating. They asked Rav Yeiva Sava to bring the wine for Birkas ha'Mazon. They reconsidered, and asked him to bring wine to drink.

Rav Yeiva: Rav taught that once one says 'let us bless', he may not drink anymore.

106a (Rabah bar bar Chanah): I was in front of R. Ami and R. Asi. A basket of fruit was brought to them. They did not give to me.

Inference: When people eat fruit together, they do not make a Zimun.

Rishonim

Rif (35a): We learn from Shmuel that whenever if they could bring something and the first two could eat from it, a newcomer joins with them. This is the Halachah.

Rambam (Hilchos Berachos 5:9): If two finished eating and a third came and ate, if the first two could eat anything with him, even other foods, they join with him.

Rosh (7:18): The Halachah follows Shmuel. Rav did not argue with him. Therefore, if two ate, and later a third came before they said 'Hav Lan u'Nevarech', he joins with them.

Ran (Chulin 29a DH ud'Amrinan): Rav Yeiva said that once one says 'let us bless', he may not drink anymore, i.e. without blessing. Rashi, R. Chananel and the Rif explain like this.

Rashi (47a DH Achilna): Rav meant 'since we finished, you cannot obligate us to make a Zimun.'

Tosfos (42a DH Tekef): Since one must bless right after washing, once one washed Mayim Acharonim, he may not eat until he blesses. The same applies if he said Hav Lan u'Nevarech, even if he did not wash.

Rashba (Teshuvah 7:56, cited in Beis Yosef OC 197 DH Kasuv): R. Ami and R. Asi were eating Peros, and they did not give to Rabah bar bar Chanah. The Gemara inferred that we do not make a Zimun on Peros. Tashbatz (Katan 304) derived that if two ate (bread) and a third came, they are obligated to give to him, so they can be Mezamen with them.

Poskim

Shulchan Aruch (OC 197:1): If two ate together and finished, and a third came, if they could bring something and the first two could eat from it, he joins with them. They are obligated to give to him to eat so he can join with them.

Eliyahu Rabah (1): The Levush says that it depends on whether one would desire to eat more. The Drishah (1) disagrees, for even one who is very satiated desires to eat a tasty food. Rather, it depends on whether one may eat, e.g. he did not yet say Hav Lan u'Nevarech. Semag explicitly says that it depends on whether he would desire to eat, like the Levush. Bahag connotes like this, and surely the Rif and Shulchan Aruch agree.

Mishnah Berurah (2): This is when the first two are not so satiated, and if appetizing foods were brought for dessert, e.g. fruits, mushrooms, chicks, or similar things, they could eat, even if only a little.

Mishnah Berurah (3): It is considered as if they did not finish their meal yet. This is if the third fixed with them, i.e. he ate at the same table with them. Ma'amar Mordechai says that the same applies if one finished eating and two came and ate. He joins with them to make a Zimun if he would be able to eat if appetizing foods were brought. The same applies to joining for a Zimun with 10.

Kaf ha'Chayim (1): There is no Chiyuv when one ate and two came later, for he need not wait for both of them to finish.

Kaf ha'Chayim (3): This is even if the two did not eat anything after he came, and they did not bring foods that they would eat.

Mishnah Berurah (4): They are not truly obligated to give to him. Rather, it is a Mitzvah to give to him, like it says above (193:1).

Kaf ha'Chayim (4): In Siman 479:1, we say that on Leil Pesach it is a Mitzvah to pursue a Zimun. This connotes that at other times, it is not a Mitzvah! There, it is a Mitzvah before the meal to pursue a Zimun. At other times, only if a third comes during the meal, one must give to him.

Bi'ur Halachah (DH Matzu): It is only if both of them could eat. If one of them washed his hands (or absolutely diverted his mind from eating anything else), the third cannot join. Eliyahu Rabah held that this is the Halachah, but then he had some Safek about this. The Chavos Ya'ir's text of the Rif supports this. Ma'amar Mordechai wrote clearly like this, and did not understand why Eliyahu Rabah was unsure.

Note: Our text of the Rif says 'if they would bring mushrooms and young fowl for you, wouldn't we eat?!' Chavos Ya'ir says that the text should read 'mushrooms for me and...', like our text of the Gemara. 'Wouldn't we eat' supports this. However, it is surprising to me that Shmuel mentioned himself before Rav.

Bi'ur Halachah (DH Mitztaref): (If the third ate,) they are obligated in a Zimun. It is not merely Reshus, like the Mechaber said above (193:2), that it depends on the end of the meal, and this is considered the end of the meal. Many Poskim hold like this, unlike R. Yonah, who says that our Sugya only discusses Reshus. Rashi (47a DH Achilna) connotes they are obligated.

Shulchan Aruch (ibid.): This is if he came before they said 'Hav Lan u'Nevarech.' If he came after they said this, he does not join with them.

Beis Yosef (DH Shnayim): Tosfos says that he joins unless they said Hav Lan u'Nevarech, for then they diverted their minds from eating.

Magen Avraham (2): According to the Ran, they join as long as they did not wash.

Bi'ur Halachah (DH Aval): The Magen Avraham says that according to the opinion that Hav Lan u'Nevarech is not a total Hesech ha'Da'as regarding eating, here they join as long as they did not wash. Several Acharonim connote like this. However, Shulchan Aruch ha'Rav says that this law is according to everyone. The Pri Megadim leaned to agree, and concluded that this requires investigation.

Kaf ha'Chayim (6): If they said Hav Lan u'Nevarech, (since it is a Safek), it is better to be passive and not make a Zimun.

Bi'ur Halachah (DH Eino): The Or Zaru'a wants to say that if three ate and they were obligated to make a Zimun, and another came and ate, even if the three washed (Mayim Acharonim) or said Hav Lan u'Nevarech, he joins with them. The other Rishonim explained differently. It seems that they disagree.

Bi'ur Halachah (DH Imahem): The Pri Megadim sides to say that whenever there is a Safek about a Zimun of three, they join (it is Reshus). To mention Hash-m's name with 10, he was unsure if it is forbidden due to the severity of Lo Sisa (saying His Name in vain). Also the Chayei Adam leans to say so.

Rema: Mayim Acharonim is like 'Hav Lan u'Nevarech'.

Bach (1): The Mordechai says that also if they washed (Mayim Acharonim), a third cannot join to make a Zimun with three. However, it seems that it is a Mitzvah for them to resume eating and wash and bless ha'Motzi, for they may do so (179:1), and the third will join with them to make a Zimun.

Magen Avraham (3): Netilas Yadayim is more (an end of the meal) than said Hav Lan u'Nevarech! Rather, the Rema teaches that if he wants to bless and eat, he may. Even though [normally] this is forbidden (179:1), here it is permitted for the Mitzvah of a Zimun. Some always permit to bless and eat. The Bach's opinion is primary.

Eliyahu Rabah (4): The Rema teaches that if he washed, all agree that a third does not join. Alternatively, he holds like the Bach, that even after Hav Lan u'Nevarech it is a Mitzvah to eat more to enable a Zimun, and he equates Hav Lan u'Nevarech to Mayim Acharonim. In order to enable a Zimun, we rely on the Poskim who permit to bless and eat. However, I found no support for the Bach's law. Presumably, since he must bless before eating, it is like a new eating. If they eat now (enough to make a Zimun on), it is like a Zimun on the new eating.

Kaf ha'Chayim (11): Since it is not a real Chiyuv to eat more, it is best to be passive (and not make a Zimun).

Kaf ha'Chayim (8): If one took a cup to bless, even though he did not say Hav Lan u'Nevarech and he did not wash, a third does not join.

Shulchan Aruch (196:3): If two Kohanim are careful not to eat bread of Nochrim, and they ate Terumah, and a Yisrael ate bread of Nochrim, they do not join for a Zimun.

Taz (196:2): If bread of Yisre'elim is available, even if they did not eat from it, they could eat from it, so they join. The Beis Yosef asked from the Halachah that whenever if they would bring something and they could eat from it, they join. Also here, this reasoning enables a Zimun!