In terms of halacha, the foremost Ashkenazic rishonim (Rema, following Maharam and Kol Bo, Maharshal and the Levush) ruled that one should protest a woman wearing tefillin (mainly due to cleanliness concerns).
However, none of the Sephardic rishonim ruled likewise. Tefillin, like any positive time-bound mitzvah, is theoretically optional if they wish to partake in it. See, for example, the Rambam (Hil. Tzitzit 3:9):
ונשים ועבדי' שרצו להתעטף בציצית מתעטפים בלא ברכה, וכן שאר מצות עשה שהנשים פטורות מהן אם רצו לעשות אותן בלא מ ברכה אין ממחין בידן, טומטום ואנדרוגינוס חייבין בכולן מספק לפיכך אין מברכין אלא עושין בלא ברכה.
Women and servants who wish to wrap themselves in tzitzit may do so without reciting a blessing. Similarly, regarding the other positive commandments which women are not required to fulfill, if they desire to fulfill them without reciting a blessing, they should not be prevented from doing so.
Here the Rambam states the opposite of the Askenazic rishonim, as one shouldn't protest women performing such mitzvot. Similarly, there is no resistance from the Shulchan Arukh about women wearing tefillin. All that he says (in OC 38:3) is:
נשים ועבדים פטורים מתפילין, מפני שהוא מצות עשה שהזמן גרמא
Women and slave are exempt from donning phylacteries for it is a
positive time-bound commandment.
While one might expect Sephardim to lead the way for women wearing tefillin, there is another twist in the halachic puzzle. Regarding mitzvot that women are not obligated to perform, the Shulchan Arukh states that women should not say a blessing before performing it, while the Rema states that they can (OC 589:6):
אע"פ שנשים פטורות, יכולות לתקוע; וכן אחר שיצא כבר, יכול לתקוע להוציאן, אבל אין מברכות ולא יברכו להן. הגה: והמנהג שהנשים מברכות על מצות עשה שהזמן גרמא על כן גם כאן תברכנה לעצמן, אבל אחרים לא יברכו להן אם כבר יצאו ואין תוקעין רק לנשים, אבל אם תוקעין לאיש המחוייב, מברכין לו אע"פ שכבר יצאו, כמו שנתבאר סימן תקפ"ה סעיף ב' הגהה א' ד"ע
Even though women are exempt, they may blow the Shofar. Similarly,
after a man fulfills the mitzvah himself, he can blow to cover the
women. However, they (the women) do not make the blessing and he
shouldn’t make the blessing for them. Rema: The [Ashkenazic]
custom is that women make blessings on a positive time-bound mitzvot,
and so in this case they should make the blessing for themselves. But
others shouldn’t make the blessing for them if they already fulfilled
the commandment. If you are not only blowing the Shofer for women, but
you are also blowing it to cover a man, you make a blessing even if
you yourself already fulfilled the commandment; as explained in 585:2.
Many Sephardic poskim (such as R. Ovadia Yosef) have followed this ruling for all such optional mitzvot (i.e., shofar, lulav, etc.) and women performing them. So we have a strange case here, where a Sephardic model might provide room for women wearing tefillin but without a beracha, and an Ashkenazic model which allows for saying brachot for optional mitzvot but doesn't allow tefillin specifically. All acharonim and modern rabbis should be taken into account (especially if one is seeking a practical solution), but it is noteworthy that all Sephardic rishonim permit tefillin wearing, like all positive time-bound mitzvot, for women to perform.
