If an Israeli resident visits the Diaspora between 7 Marheshven and the beginning of December, the period in which Israelis recite "Barech Alenu" in the Amida while Diaspora communities continue reciting "Barechenu," he follows the custom in Israel. Even though he is currently in the Diaspora, he recites "Barech Alenu" just as he would in Israel. (This is assuming that he has come to the Diaspora for a temporary visit, and not for an extended stay.)
Hacham Ovadia Yosef, in his work Yehave Da’at (vol. 2, end of Siman 11; listen to audio recording for precise citation), addresses the case of an Israeli resident in this situation who forgets to recite "Barech Alenu." In this case, Hacham Ovadia writes, he does not repeat the Amida. Even though he should ideally recite "Barech Alenu," in accordance with the practice in Israel, nevertheless, since he currently finds himself in the Diaspora, he does not repeat the Amida if he mistakenly recited "Barechenu."
With regard to the reverse case, where a Diaspora resident temporarily visits Israel during this period, Hacham Ovadia rules that he should recite "Barech Alenu" during his stay in Israel. It is very common for people to travel to Israel from the U.S. at the end of November. Visitors during this period must be aware that they should recite "Barech Alenu" instead of "Barechenu" while they are in Israel. Upon their return to the Diaspora, Hacham Ovadia rules, they resume the recitation of "Barechenu" (assuming they return before the beginning of December, when Diaspora communities begin reciting "Barech Alenu"). This case thus gives rise to an interesting situation, whereby one recites "Barech Alenu" for a brief period and then reverts back to "Barechenu." Hacham David Yosef writes (listen to audio recording for precise citation) that in order for this not to appear contradictory, a person in this situation should, upon returning to the Diaspora, recite the words "Ve’ten Tal U’matar Li’bracha" in the Beracha of Shema Kolenu. Since he had begun reciting "Barech Alenu" during his stay in Israel, it would be proper for him to at least recite the words "Ve’ten Tal U’matar Li’bracha" so that he continues the special prayer for rain even after returning to America.
Summary: If an Israeli resident visits the Diaspora at some point between 7 Marheshvan and the beginning of December, he recites "Barech Alenu" in the Amida, as is done in Israel during this period. Nevertheless, if he mistakenly recited "Barechenu," he does not repeat the Amida. Diaspora residents who visit Israel during this period should recite "Barech Alenu" while in Israel, and if they return to the Diaspora before the beginning of December, they resume reciting "Barechenu." Preferably, in such a case one should recite the words, "Ve’ten Tal U’matar Li’bracha" in the Beracha of "Shema Kolenu."
