One of the reasons brought down for washing Mayim Achronim before reciting birchas hamazon is that one should not bentch with dirty hands (Mishna Berurah 181:1). Since it is likely that ones hands became dirty during the course of a meal one must wash mayim achronim to remove that dirt before reciting birchas hamazon.
The Mishna Berurah writes further that if during a meal, an “Istonis” (someone who is extra careful to always be clean) needs to make a bracha (e.g. wine, dessert), he must wash his hands prior to making the bracha being that his hands are probably dirty from touching food during the meal (181:23). This seems to be an extension to the halacha of mayim achronim in that before reciting any bracha an Istonis must make sure his hands are clean.
The Piskei Teshuvos (181:3) brings down a Kaf HaChaim (181:5) which says that this halacha applies to everyone and not just an Istonis. Therefore, one must always be careful to have clean hands before making any bracha. This applies during a meal or even when eating food not during a meal. Before making any bracha a person must make sure his hands are clean, and if they are not he must first wash his hands and then make the bracha.