If two people ate a meal, and after they washed Mayim Aharonim in preparation for Birkat Ha’mazon, a third person joins them and eats, they cannot recite a Zimun. Once the first two washed Mayim Aharonim, they established the end of the meal, such that the arrival of a third person cannot facilitate the formation of a Zimun. Moreover, even if the first two had not yet washed Mayim Aharonim, but merely said, "Hab Lan Ve’nibrich" ("Come, let’s bless"), expressing their intent to now recite Birkat Ha’mazon, this marks the conclusion of the meal, such that a Zimun is not recited even if they are then joined by a third person.
If they finished eating but had not yet expressed their intent to recite Birkat Ha’mazon, and at that point they are joined by a third person who then eats, they form a Zimun, even though the first two already finished eating. However, this applies only if the first two would eat if more food was brought to them. If they are already fully satiated, and no longer have any interest in eating, then they cannot form a Zimun with the third who arrives and eats after they finished eating.
What must the third individual eat in order for the three men to form a Zimun? Must he eat a Ke’zayit of bread, like the first two, or does it suffice if the third person eats other foods?
The Shulhan Aruch rules that when ten people eat together, and the issue at stake is whether the Name of G-d ("Elokenu") can be recited in the Zimun due to the presence of a Minyan, it suffices if even just seven members of the group ate bread, and three ate other foods. As long as seven people ate a Ke’zayit of bread and three ate a Ke’zayit of other foods, a Zimun is recited with G-d’s Name.
The more complex question is whether three people can form a Zimun if only two people ate bread and the third ate other foods. The Shulhan Aruch first establishes that all three must eat bread for a Zimun to be recited, and then proceeds to cite an opinion that it suffices for the third person to eat a "Mezonot" food, and a third opinion that it suffices for the third to eat any food, even fruits of vegetables. In light of this debate, the Shulhan Aruch writes that if two people ate together, and then a third person arrives, they should offer him bread. If he accepts the offer and eats bread, then they form a Zimun, but if he does not want bread, then the other two should not offer him other food, in order to avoid this Halachic question.
However, the Kenesset Ha’gedola (Rav Haim Benvenisti, 1603-1673) writes that the accepted practice among Sepharadim was to rely on the lenient position, and recite a Zimun even if the third member of the group ate only fruits and vegetables. This is the practice that is followed today. And thus, if two people who ate bread are joined by a third, although it is preferable for him to eat bread, the three people form a Zimun even if the third person eats a Ke’zayit of other foods.
The Poskim rule that even if the third person ate nothing, the three can form a Zimun if he drank a Rebi’it of certain beverages – beverages such as wine, beer and natural fruit juices. It does not suffice for the third to drink water or flavored water, such as tea, coffee, or fruit juice made from concentrate. Hacham Bension Abba Shaul (Israel, 1924-1998) maintained that soft drinks are considered flavored water in this regard. Seltzer, too, is to be treated like water with respect to this Halacha. (Although the Aruch Ha’shulhan suggested that seltzer is not considered water, the consensus does not follow this opinion.)
Summary: If two people ate a meal, and they were joined by a third who came to eat after they washed Mayim Aharonim, or after they said, "Let’s recite Birkat Ha’mazon," they do not recite a Zimun. If he joined them earlier, then even though the first two had already finished eating, they form a Zimun, unless the first two are fully satiated and would not eat more food if they were offered. The three form a Zimun even if the third person did not eat bread, as long as he ate a Ke’zayit or other foods or a Rebi’it of beverages such as wine, beer or natural fruit juices (as opposed to water, tea, coffee, soft drinks, or concentrate-based juice).