I switched areas during my meal taking my food with me, with prior intent to do so by Hamotzi, and forgot to eat a piece of bread while I was in my second area and only remembered after I already said Al Naharos Bavel and performed Mayim Achronim? What do I do now? Can I still eat a piece of bread and if so do I have to say a new blessing over it?
You should wash without a blessing, say Hamotzi and eat any amount of bread, repeat Mayim Achronim, and then Bentch.
Explanation: Since you may only recite Birchas Hamazon in the last area that you ate bread, you must therefore eat bread in your current area in order to be allowed to Bentch there. Now, since you already washed Mayim Achronim therefore a new blessing of Hamotzi must be recited over the bread even if you plan to only eat a tiny amount. Likewise, since you need to repeat the blessing of Hamotzi over the bread therefore you must also wash your hands again for bread but you should only say a blessing if you plan on eating more than a Kebeitza of bread. Now, since you made a Hefsek after your Mayim Achronim, therefore you must repeat the Mayim Achronim. Now, although there are Poskim who rule that the custom is to never eat or drink after Mayim Achronim, Admur omitted this opinion in his Seder Birchas Hanehnin, and likewise their opinion would not apply in this case.
Sources: See regarding repeating the blessing of Hamotzi if did Mayim Achronim: Admur 179:5; 477:7; Seder Birchas Hanehnin 5:5; Luach 7:5; Michaber 179:1; Brachos 42a; M”B 179:2; Shulchan Hatahor 179:3; Chayeh Adam 59:8; Ketzos Hashulchan 42:7; Piskeiy Teshuvos 179:2 See regarding that must re-wash if need repeat Hamotzi: Seder Birchas Hanehnin 5:6; Admur 179:5; M”A 179:2; Biur Halacha 178:2; Piskeiy Teshuvos 184:2 See regarding repeating Mayim Achronim if make Hefsek in between: Admur 179:5; 271:12; Seder Birchas Hanehnin 5:5; Luach 7:5; Taz 179:1; M”A 179:2; See regarding the obligation to recite Birchas Hamazon in the area in which one eats the bread: Admur 184:1; 178:4; Seder Birchas Hanehnin 9:16; Luach Birchas Hanehnin 6:16; Michaber 184:1; See regarding that eating any amount of bread suffices for this purpose: Admur 184:1 in length that in all cases a small amount of bread suffices even less than Kezayis and explains reason why; 178:4 [eats bread there…eats a little in his second area]; Seder Birchas Hanehnin 9:16 and Luach Birchas Hanehnin 6:16 [makes no mention of amount of bread to eat and says simply bread]; Michaber 184:2 “A little [bread]”; M”A 184:4; Lechem Chamudos 34, brought in Beir Heiytiv 184:4; Kneses Hagedola 184; Chayeh Adam 184:9;M”B 184:8-9; See Kaf Hachaim 184:10; Piskeiy Teshuvos 184:1 footnote 4 Stringent to eat Kezayis: Halachos Ketzanos 1:147, brought in Beir Heiytiv 184:4 and Yad Aaron on Tur and Kaf Hachaim ibid and Piskeiy Teshuvos ibid