The Maharam of Rotenberg (Germany, 1215-1293) was of the opinion that Tisha B'Av, the saddest day of the year, has the same status in Halacha as the first day of mourning, and thus just as one does not wear Tefillin on his first day of mourning, Heaven forbid, so do we all refrain from wearing Tefillin on Tisha B'Av.
Other authorities, however, held that on Tisha B'Av one must, indeed, wear Tefillin, and this is the accepted position.
There is, however, considerable controversy regarding the question of when one wears Tefillin on Tisha B'Av. The Gaon of Vilna (1720-1797) had the practice of donning Tefillin only at midday on Tisha B'Av. He apparently held that the first half of Tisha B'Av was a period of intense mourning, whereas the second half the day, when we begin to recite "Nachem," marks a period of lesser intensity and thus warrants wearing Tefillin. This is indeed the practice of many Ashkenazim. Other Ashkenazim have the practice to don Tefillin on Tisha B'Av only during the Mincha service.
What is the practice of the Sepharadim in this regard?
From numerous sources, including the Kaf Ha'chayim, Ben Ish Chai and writings of the Kabbalists, it emerges that the Sephardic practice is to wear Tefillin on Tisha B'Av during Shacharit in the morning, as on all other days, with the standard Beracha. People with the practice of wearing Tefillin Rabbenu Tam every morning should do so on Tisha B'Av morning, as well.
We do, however, find divergent customs as to where the Tefillin should be worn. The Kaf Ha'chayim and Ben Ish Chai mention that on Tisha B'Av one dons Tefillin and recites Shacharit privately, at home, and then comes to the synagogue after removing his Tallit and Tefillin for the repetition of Shemona Esrei, Kinot and Eicha. Others, however, held that one prays on Tisha B'Av in the synagogue with Tallit and Tefillin as usual, just as on other days, and the Tallit and Tefillin are removed after the repetition of Shemona Esrei. This was the ruling of Chacham Baruch Ben-Chaim. Both practices have roots in Halachic literature, and thus each community should follow the guidance of its Rabbi. It must be emphasized that according to all authorities, one does not wear Tefillin which serves as a sign of glory and royalty while reciting Kinot to lament the destruction of the Beit Ha'mikdash.
Summary: Tefillin are worn on Tisha B'Av with the Beracha. Ashkenazim have the custom of wearing Tefillin on Tisha B'Av only in the afternoon, whereas Sepharadim wear Tefillin already in the morning, as usual. Some Sepharadim wear Tefillin and pray Shacharit privately at home and then come to synagogue without Tefillin, whereas others pray with Tallit and Tefillin in the synagogue as usual. Each community should follow its custom as instructed by its Rabbi.
