The Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909) discusses a situation of two people who ate a meal together, and had washed Mayim Aharonim in preparation for Birkat Hamazon. Of course, when only two people eat together, they cannot recite a Zimun before Birkat Hamazon. However, the Ben Ish Hai writes, if a third person joined them before they recited Birkat Hamazon, they should eat a new meal in order to facilitate a Zimun. Even though Halacha generally requires one to recite Birkat Hamazon immediately after washing Mayim Aharonim, an exception is made in this instance due to the importance of Zimun. The two men should wash their hands again, recite a new Beracha of Hamosi and eat some bread, together with the third person who had just arrived. They should then recite Birkat Hamazon with a Zimun. According to the Ben Ish Hai, reciting a Zimun is of such great importance that we require these two individuals to recite extra Berachot and eat a new meal in order that they may recite a Zimun before Birkat Hamazon.
Hacham Ovadia Yosef, in his Halichot Olam, disputes the Ben Ish Hai’s ruling. In his view, once the two men washed Mayim Aharonim they must recite Birkat Hamazon, even at the expense of a Zimun.
Nevertheless, the Ben Ish Hai’s ruling – even if other authorities disagree – demonstrates the great importance of a Zimun. The Ben Ish Hai writes that before a person recites Birkat Hamazon, there are certain Mekatregim (harmful spiritual forces) that threaten him, and the Zimun has the effect of clearing away these forces. We must therefore not underestimate the critical importance of the Zimun, and we should all approach it with utmost seriousness and reverence.
Summary: If two people ate together and washed Mayim Aharonim in preparation for Birkat Hamazon, they must recite Birkat Hamazon immediately, and even if a third person arrives, they should not eat again in order to recite a Zimun. This is the accepted Halacha, though some authorities require the two individuals to eat more bread in this case so that they could recite a Zimun, demonstrating the importance that Halacha affords the Zimun.