Siman 80

One who is unable to restrain himself from flatulence because he flatulates against his will, and it is impossible for him to complete the recitation of Shema and the prayer without flatulence—it is better that the time for the recitation of Shema and the prayer should pass than that he should pray without a clean body. And if the time for prayer passed—he is under duress, and he prays the Mincha prayer twice. And if it appears to him that he can restrain himself during the time of the recitation of Shema—he should don tzitzit between "Ahavah" and the recitation of Shema, and recite the blessing over them. Thus ruled the Tur and the Shulchan Aruch, and its source is from a responsum of the Rosh (beginning of Kelal 4, see there).

And there is one who wrote that it does not appear so according to the halacha (the Gra). And there is one who wrote in this wording: Even though in the Gemara we only say "one who needs to relieve himself shall not pray," this is because it seeks to say: "and if he prayed, his prayer is an abomination." But it is also true that regarding flatulence as well, it is better that he not pray now than to pray with an unclean body… And all this is when an incident occurred to him due to an upset stomach from some food, in a manner that by evening he will certainly be able to pray with a clean body. But if he has an illness because of which he flatulates at all times – he shall pray, and when the flatulence comes to him – he shall remain silent, and wait until the odor dissipates. And if he tarries long enough to complete the whole of it – he shall return to the beginning (Pri Chadash).

Our teacher, the Rema, wrote in a responsa (siman 98): Since he did not say that there is a prohibition in the matter, but rather it is better for him to wait since he has a remedy to pray two during mincha. However, one who is accustomed to this, and if he does not pray he will be exempt from prayer forever – it is permitted for him to pray without a clean body. For a rabbinic prohibition does not come and push aside a Torah obligation, or prayer which is a mitzvah of the public. Thus far his words. And this is similar to what I wrote in the previous section. And in truth, the words of the Tur and Shulchan Aruch are not understood: for granted that for prayer a completion-prayer is effective, but regarding shema there are opinions whether a completion-prayer is effective, as I wrote at the end of siman 58, see there (Eliyah Rabbah). And it is even more puzzling that in the responsa of the Rosh he did not mention shema at all, only prayer, see there (ibid.).

Therefore, there is one of the great authorities who ruled that in truth, only the prayer he should not pray. And likewise, he should not say Pesukei D'Zimra, which were instituted because of the prayer. However, he should say Shema even without a clean body, for prayer is like standing before the King, which is not the case with Shema. And even if he is unable to don tefillin—he should recite Shema without tefillin (the HaGra"z). And he may also say the blessings of Shema. And if he can don tefillin between "Ahavah" and Shema—he should don them with a blessing. And regarding the tallit, it is obvious that he can don it with a blessing, for it is permitted to enter a latrine in a tallit (Taz). And so one should instruct. And also that which has been explained, that if he has an illness from which he passes gas, he should certainly pray as well—it is a clear law, and such is the halacha. (And also the Tur and Shulchan Aruch who wrote: It is better that the time for Shema and prayer pass and he should not pray...—their intention was only regarding prayer, for they did not write "and he should not recite and should not pray" but rather "and he should not pray." And that which they wrote regarding the time for Shema and prayer, meaning: that he should not pray even if the time for Shema passes, and even if the time for prayer passes. Examine carefully and you will find it simple. And he must recite the morning blessings.)