from the rishonim it is clear one should finish neila by tses the shagas aryeh and magen avraham say ein limchos if a tzibur cont. neila after tzes. where did the minhag/halacha originate that we may cont. neila after tzes thanks and kesiva vchasima tova.
Answer:
The Mishnah Berurah notes that “there are those to rely upon” for davening Ne’ilah after Tzeis, and this is probably a reference to those Rishonim who write that one can continue into nightfall: The Roke’ach, Riaz and Ravyah, who rule like Rav, namely that one who davnes Ne’ilah does not have to daven Maariv.
The assumption is that according to Rav Ne’ilah is a “night prayer” and therefore it takes the place of Maariv.
Although the Shulchan Aruch does not rule this way, we can rely on them when time is tight (though the general custom today is not to do so).
Best wishes.
