When a group of three men eat a bread meal together they are required to make a zimun for bentching. When a group of ten men eat together at the same type of meal, they form a zimun and can use the shem Hashem when reciting the zimun.
In instances where a zimun of three is one man short or a zimun of ten is short three men or fewer, the zimun may be completed with people who did not eat bread. While the Piskei Teshuvos (197:2) writes that it is preferable that the person who completes the zimun should have eaten food on which a mezonos is made, such a preference is not required. The Shulchan Aruch (197:2) writes that even if the third man merely ate vegetables or drank a reviis of anything other than water, he may be counted towards the zimun if he is required to make a bracha achrona. However, the Mishna Brura warns (197:9) that the person who did not eat bread may not recite a bracha achrona (al hamichya/borei nefashos) prior to joining the zimun. Once a bracha achrona is recited, he may no longer join the zimun.
This halacha is especially relevant on Pesach, a time when there are no mezonos products available at least for those who do not eat gebruchts. Even when the third person didn't eat a mezonos product he can still join the zimun if he ate any food other than water which requires a bracha achrona.
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