Historically, ideological matters were kept separate from halakhah. The various rabbinic organizations in Europe and the United States were comprised of rabbis who held different views about Zionism and other matters. If you were a Mizrachi supporter but the rav of your town was an Agudist, you still asked him all of your halakhic questions, because he was your rav. By the same token, if you were an Agudist and the rav of the town was Mizrachi, he was still the one to answer your halakhic questions. This is how matters worked in Lithuania and Poland and then in the United States. One of the unfortunate results of modern haredi Judaism (and it has precedents in Germany and Hungary) is that this model was destroyed.
Marc B. Shapiro, “Some Unusually ‘Liberal’ Statements by Mainstream Rabbinic Figures”, The Seforim Blog (9 February 2016) [http://seforim.blogspot.com/2016/02/some-unusually-liberal-statements-by.html]