It is obligatory to wash the hands after eating, before reciting Birchas HaMazon, even if the hands are clean. This washing is known as “Mayim Achronim” (Shulchan Aruch Siman 181:1. See Mishna Berura ibid. S”K 1 regarding where Chazal derived this Halacha and some of its reasons)
Although the Shulchan Aruch quotes a more lenient opinion about this (in Siman 181:10), the Poskim are extremely stringent with this and dismiss the leniencies even nowadays, based on many deep kabalistic reasons, and implore everyone to be careful in the observance of Mayim Achronim (See Mishna Berura 181:21, Shl”a Sha’ar HaOsiyos, Kuf, Aruch HaShulchan 181:3 and 4, Kitzur Shulchan Aruch Siman 44:1. The Kaf HaChaim 181:27 writes that one that is lenient with Mayim Achronim, risks having his life shortened.)
Even those that are not careful with Mayim Achronim when eating with others, should be extra careful when eating alone, as the impure spirits can cause much more harm when one eats alone. (Kaf HaChaim 181:1. See also the piercing words of the Pele Yoetz in “Netilas Yadayim”)
Even those who are not so careful with Mayim Achronim during the weekdays, should be extra stringent with this on Shabbos and Yom Tov. (Leket Yosher in Hilchos Seudah, quoting the Terumas HaDeshen. See also Mateh Efraim Siman 583:4 and Siman 619:1 and in the Elef L’mateh ibid.)