If two men ate a meal together, they obviously do not recite a Zimun, which is recited only when three or more people at together. The question arises, however, whether a Zimun is recited in a case where a third person comes and joins these two people after they had finished eating, but before they recited Birkat Hamazon. If this third person now eats something, can he be considered part of the meal eaten by the first two men, such that the three of them can recite a Zimun?
The Halacha in this case is that the three men recite a Zimun if all the following conditions are met:
1) The first two men are prepared to wait for the third person to finish eating. If they are rushing and do not want to wait, then they recite Birkat Hamazon without a Zimun.
2) The first two men had not washed Mayim Aharonim and did not even say to one another, "Let us say Birkat Hamazon." Once they made either of these preparations for Birkat Hamazon, their meal is considered finished, and it is thus too late for a Zimun even if a third person joins before the actual recitation of Birkat Hamazon.
3) They would be able to eat a food that they enjoy if it would be served. If they have eaten to the point where eating anything else would constitute "Achila Gasa" (gluttonous eating), then they cannot recite a Zimun with the third man, as their meal is considered finished. If, however, they would still have room to eat a food they enjoy, then they may be regarded as having yet to complete their meal, and may thus join with the third man to recite a Zimun.
