It is customary on a fast day, such as Shiba Asar Ba’Tammuz, to wear Tallit and Tefillin during Minha (if one is still fasting). The Bet Yosef (commentary to the Tur by Maran, author of the Shulhan Aruch), in Orah Haim (46), writes that this is done in order to give us the opportunity to recite an additional two Berachot (one over the Tallit, and one over the Tefillin). Halacha requires reciting 100 Berachot each day, and on a fast day, when we do not recite Berachot before or after eating or drinking, we risk falling short of the required 100 Berachot. The Bet Yosef presents a detailed calculation showing how these two extra Berachot over the Tallit and Tefillin at Minha could make the difference and help us reach 100.
Which Tefillin should one wear during Minha on a fast day?
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that the Arizal (Rav Yishak Luria of Safed, 1534-1572) used to wear Tefillin Rabbenu Ram during Minha every day (and not only on fast days). During Shaharit, he would wear Tefillin Rashi and Tefillin Rabbenu Tam together, and at Minha, he would wear only Tefillin Rabbenu Tam. Later, however, the Arizal changed his practice, and began wearing at Minha the special kind of Tefillin called "Shimusha Rabba," which are Tefillin Rashi, but exceptionally large.
Accordingly, Hacham Bension Abba Shaul (Israel, 1924-1998) writes that at Minha on fast days, one who has a pair of "Shimusha Rabba" should wear this Tefillin, and recite a Beracha. He disagrees in this regard with Hacham Ovadia Hedaya (1889-1969), who writes in his work Yaskil Abdi that the "Shimusha Rabba" is on such an exalted level that we cannot connect to this kind of Tefillin nowadays, and therefore no Beracha should be recited when putting on such Tefillin. The purpose of the Beracha, Hacham Ovadia Hedaya explains, is to bring the spiritual impact of the Misva down into one’s soul, and in the case of the "Shimusha Rabba," this spiritual impact is beyond our capability. As such, the Beracha over this Misva would be a Beracha Le’batala (Beracha recited in vain). He thus ruled that one who wears this Tefillin does not recite a Beracha. Hacham Bension agrees that the power of the "Shimusha Rabba" lies beyond our limited abilities nowadays, but in his view, this does not affect the Beracha, and therefore one who wears "Shimusha Rabba" indeed recites a Beracha.
Hence, if one has "Shimusha Rabba," he should wear this Tefillin at Minha on fast days, with a Beracha. Even if no one else in the synagogue wears such Tefillin, he may wear it, and this does not violate the prohibition of "Lo Titgodedu" (deviating from the communal practice).
If one does not have "Shimusha Rabba," then at Minha on fast days he wears Tefillin Rabbenu Tam. This is the conclusion of the Hid"a (Rav Haim Yosef David Azulai, 1724-1806), after a lengthy discussion in his Mahazik Beracha (Orah Haim, 1).
However, this conclusion gives rise to a difficult question. The accepted practice is not to recite a Beracha over Tefillin Rabbenu Tam, and thus when wearing Tefillin Rabbenu Tam at Minha on fast days, no Beracha is recited. But as we have seen, the entire reason for wearing Tefillin at Minha on a fast day is to add more Berachot. Seemingly, then, wearing Tefillin Rabbenu Tam undermines the very purpose for which Tefillin is worn at this time.
The answer is that there is also an additional reason for wearing Tefillin during Minha on fast days. The Rambam (Rav Moshe Maimonides, Spain-Egypt, 1135-1204), at the end of Hilchot Tefillin, describes the effect that wearing Tefillin has on a person’s mind and behavior. While one wears Tefillin, the Rambam writes, he is overcome by humility and fear of G-d. This is why we do not wear Tefillin the entire day as was done in yesteryear – because we would be unable to maintain this lofty mindset and spiritual focus throughout the day, and so we wear it only during Shaharit in the morning. On the afternoon of a fast day, however, we feel frail and subdued because of the fast, and so we are able to wear Tefillin and experience the special feelings of humility and subservience to Hashem which Tefillin bring.
This reason, of course, has nothing at all to do with the Beracha, and this explains the accepted practice to wear Tefillin Rabbenu Tam at Minha on fast days, even though no Beracha is recited.
Summary: It is customary to wear Tallit and Tefillin at Minha on fast days (if one is still fasting when he prays Minha). One who has "Shimusha Rabba" Tefillin should wear this Tefillin, with a Beracha, and one who does not should wear Tefillin Rabbenu Tam, without a Beracha.