The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Lech-Lecha (5), writes that one should ensure to use only a small amount of water when washing Mayim Aharonim (the washing done before Birkat Ha’mazon). He explains (listen to audio recording for precise citation) that the water of Mayim Aharonim is associated with the "Sitra Ahara" (literally, "the other side"), the forces of impurity, and we therefore should use small amounts of water in order not to strengthen the hostile spiritual forces.
The Ben Ish Hai then proceeds to address the case of a person whose hands are dirty, such that a small amount of water would not suffice to clean them. In such a case, he writes, one should thoroughly wash his hands – and, if necessary, his mouth – normally, at the sink, with the intention of continuing to eat. Since he intends to continue eating after washing, this washing will not be considered Mayim Aharonim, and thus he may use as much water as he needs to clean his hand and face. He should then return to the table, eat a morsel of bread or other food, and then wash Mayim Aharonim with a small amount of water. This way, he is able to properly clean his hands while still using only a small amount of water for Mayim Aharonim.
Summary: One should use a very small amount of water for Mayim Aharonim. If one’s hands are dirty, then he should wash them thoroughly with the intention of continuing to eat afterward. After washing, he should eat something, and then wash Mayim Aharonim and recite Birkat Ha’mazon.
