Beis Yosef Orach Chaim 34:2

What Rabbeinu (the Tur) brought in the name of the Rosh, that one should wear two pairs of tefillin—this is also written in the Sefer Mitzvos Gadol (SeMaG), the Sefer Mitzvos Katan (SeMaK), and the Sefer HaTerumah.

The SeMaG and the Sefer HaTerumah also write that if someone doesn’t know how to position them correctly and wear both pairs at once, he should first put on one set—the shel yad and shel rosh—and then remove them immediately and put on the other set, relying on the first berachah. And even though he’s interrupting between the two shel yad by putting on the shel rosh in between, this is still better than not wearing them at all.

In the Shimusha Rabba, it’s written like the order of Rashi, and the same is written in the Rambam (Hilchos Tefillin 3:5).

The SeMaG and the Mordechai write that a letter was sent from Eretz Yisrael reporting that a box fell into the sea near the kever of Yechezkel HaNavi, and they found very old tefillin inside—laid out like the seder of the Rambam and Rashi.

The Rashba, in a Teshuvah, discusses this matter at length, refuting the view of Rabbeinu Tam and upholding the position of Rashi and the Rambam. At the end of his Teshuvah, he writes:

“Bottom line, my opinion leans towards Rashi and the Rambam, and upon them we rely and follow their practice.”

Likewise, Rabbeinu Yonah and the Ramban also followed this practice themselves.

This Teshuvah can be found in the Teshuvos HaRamban, Siman 234.