How do we pasken by someone who eats a seudah then throws up afterwards do they still have to bentch bircas hamazon?
Birkas Hamazon is not recited if the meal is thrown up.
The Mishnah in Berachos (51b) writes that Birkas Hamazon is recited until the time that the food is digested by the person who ate.
This principle is ruled by the Shuchan Aruch (184:5).
The Magen Avraham explains that if a person eates a small meal, the time for digestion, and therefore the time for Birkas Hamazon, will be smaller. The Taz, however, disputes this point.
Therefore, somebody who throws up after his meal, can be compared to “after the time of digestion,” because he certainly doesn’t feel full from the meal he has thrown up. Therefore, Birkas Hamazon cannot be made.
This ruling is given explicitly by the Shaarei Teshuva (208:1) in the name of the Birkei Yosef, for the reason explained above.