What is the ruling on the exit of an impurity from the anus, less than the size of a barley grain, without leaving any trace on the clothes or any smell, during defecation?

All praise is due to Allah, and peace and blessings be upon the Messenger of Allah. As for what follows:

Impurities in the anus can be in one of two situations:

1. Persistence of impurity after Istinja' (cleansing after defecation/urination): If the impurity remained after Istinja' and was not removed, it is obligatory to remove it. There is no need to repeat the prayers you performed with this impurity present, because even though avoiding impurity is a condition for the validity of prayer, it is a condition contingent upon knowledge and ability. Therefore, the prayer of one who is unaware of the presence of impurity or is unable to remove it is valid according to the most authoritative opinion. It is important to note that some jurists (like the Hanbalis) stipulate that Istinja' must precede Wudu' (ablution) and consider Wudu' invalid if this is not done, while the majority view that Istinja' is not a condition for the validity of Wudu'.

2. Discharge of impurity after Wudu': If impurity is discharged after Wudu', then Wudu' becomes invalidated, because it is nullified by anything that exits from the two private parts. In this case, it was obligatory to perform Wudu' after removing the impurity and after the time for prayer had begun, and then perform the prayer. The prayers you performed before removing the impurity and repeating Wudu' must be repeated due to their invalidity.

If you are unsure which of the two situations applied, the criterion is your state before the doubt. If you were in a state of purity before the doubt, then you are now in a state of ritual impurity. If you were in a state of ritual impurity, then you are now in a state of purity. If you do not know your state before either of these, the default is ritual impurity, and you must purify yourself.

As for the garment, it is not obligatory to purify it unless you see traces of impurity on it or you are reasonably certain that it has become impure.