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Question:
After finishing seuda shlishit I forgot to recite Birkat Hamazon until coming back from Ma’ariv. Was I supposed to say R’tzei in Birkat Hamazon at that point?

Answer: This question tests the primacy of the following competing halachic rules. One is that once one becomes obligated in one of the additions to Birkat Hamazon, he is to recite it even after the time that it normally applies. The other is that during times that can relate to the preceding period or the following period, we do not allow there to be an internal contradiction (tartei d’satrei) and apply it to both. Let us survey sources and applications of each rule before coming to an answer.
The first rule is actually not unanimously held. The Rosh (Shut 22:6) says that if one starts seuda shlishit before sunset but bentches after the day is over, he does not recite R’tzei. To the contrary, if he ate a meal before Shabbat but recited Birkat Hamazon only on Shabbat, he would then recite R’tzei. The Tur (Orach Chayim 188) says that according to [his father] the Rosh, one who did not bentch on his Purim seuda until nightfall after Purim he would not recite Al Hanisim. In other words, we always follow the time that one is ready to make the pronouncement. However, we do not adopt the Rosh’s opinion as standard halacha. Rather, we follow the time that one ate and say that if he ate in a way that caused an obligation of a special pronouncement such as R’tzei or Al Hanisim, he is to make it even when its time ostensibly over (Shulchan Aruch, Orach Chayim 188:10). (An exception to this is Sheva Berachot that finish after the week is over, as one cannot make independent berachot after the time period- see Sha’arei Teshuva 188:8 based on the Ginat Veradim).
The next rule, that we avoid tartei d’satrei situations, has far too many applications to address, and we will stick to those close to our topic. One discussion is about one who started a seudat Purim on Erev Shabbat and continued it on Shabbat, making Kiddush in the middle. When he eventually bentches, there is reason to say both Al Hanisim, due to that which he ate during the day, and R’tzei, due to that which he ate after Kiddush. The Magen Avraham (695:9) says that one recites Al Hanisim, apparently in addition to R’tzei (see Sha’ar Hatziyun 695:19), whereas the Chayei Adam (155:32) says that you cannot say both Al Hanisim and R’tzei in the same Birkat Hamazon when the two relate to different days.
Another case where the issue of tartei d’satrei arises regarding R’tzei is when one starts eating on Shabbat and continues eating bread after night begins and Rosh Chodesh starts. There is a theoretical obligation to say both R’tzei and Ya’aleh V’yavoh but, again, the two belong to two different days. Here, the Magen Avraham (188:18) says that one cannot say both and since it is more clear that there is an obligation of Ya’aleh V’yavo than it is that there is an obligation of R’tzei, one says just Ya’aleh V’yavo (we will not deal with reconciling the Magen Avraham’s two rulings). The Taz (188:7) says that one may say both despite the apparent contradiction, and the Mishna Berura does not come to a clear distinction.
There is some logic to say that tartei d’satrei exists only when one has to say two contradictory things in the same context, e.g., within Birkat Hamazon. If so, since there is nothing in Birkat Hamazon that indicates that it is Motzaei Shabbat, saying R’tzei would not be a problem. However, the consensus of poskim is that davening Ma’ariv is such a strong indication of ending Shabbat that it is not possible to say R’tzei in bentching afterward (see implication of Beit Yosef, OC 188, Magen Avraham 188:17; Mishna Berura 188:32; Aruch Hashulchan 188:27). The loss of R’tzei is also not a critical matter. Because not all agree that one has to eat bread at seuda shlishit, if one forgets R’tzei at sedua shlishit, he does not repeat Birkat Hamazon (Shulchan Aruch, ibid. 8). What some major poskim (Magen Avraham 263:33; Rabbi Akiva Eiger to Magen Avraham 188:17) are unsure about is if one said Hamavdil but did not daven, whether that precludes saying R’tzei.