The R”T about tefilin is a tosafos in menochos. I learned it many years ago and as far as I can remember, the halocho is like Rashi. R”T’s shita is accepted as halocho b’Toras hanistar (whatever that means). Therefore people who are more ‘into’ kabalah, i.e. Chassidim and Sefardim, will wear R”T tefilin in addition to Rashi tefilin. Similarly, R”T wearers will not wear tefilin on chol hamoed, when the toras hanistar psak is that one is chayav misa for wearing tfilin.
Many non-chasidish, non-sefardi people have taken on wearing R”T as an added hiddur.
One more point, one who wears Rashi and R”T will also be yoitze the other two shitos, as wearing the parshiyos in reverse order is also yoitze. That’s why a righty who wears lefty tefillin is yoitze.