Hello99 wrote: “I clarified with my Erloi neighbor the conduct in the Erloi Beis Medrash. In Erloi in Beitar tefillin are only put on at home before davening and not in shul at all. In Yerusholaim, the Rebbe has always worn tefillin in the Ezras Nashim or Sukka with a minyan of just 10 men accompanying him.”
However, the door between the Minyonim is always open (between the Sukka and the Beis Medrash) and the chazan (wearing Tefilin) for Shacharis and Hallel stands in the doorway as chazan for both Minyonim.