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Ibn al-Hajj al-Fasy (maliki) said in his al-Madkhal المدخل (My own translation take it carefully):

Ibn al-Haaj al-Fasy (maliki) said in his al-Madkhal المدخل (My own translation take it carefully):

First of all who told you that you should read tasbih after the 4th raka' of Taraweeh?

I was astonished that I've never heard about this until reading your post!

Any act in Islam needs a clear backup but scholars have three different opinions on this some said it is permissible other not and others even said it is bida'h among those who said it is permissible you may find scholars like al-Kassani in his book Badai' as-Sanai' بدائع الصنائع from the hanafi school of fiqh and al-Mirdawi -Hanbali scholar- in his al-Insaf الإنصاف . Here the statement of al-Kassani (My own translation take it carefully):

Ibn al-Humam (Hanafi) in his Fath al-Qadir فتح القدير only confirms the sitting or pausing between two sets of taraweeh as a custom of the people of the two harams (Mekka and Medina). So what people have done in between was up to them and that they make tasbeeh etc. is possible option. He also made clear that people of Medina used to pray during this 4 optional raka'a's.

I was astonished that I've never heard about this until reading your post!

Any act in Islam needs a clear backup, but in this case scholars have three different opinions on the matter some said it is permissible other not and others even said it is bida'h among those who said it is permissible you may find scholars like al-Kassani in his book Badai' as-Sanai' بدائع الصنائع from the hanafi school of fiqh and al-Mirdawi -Hanbali scholar- in his al-Insaf الإنصاف . Here the statement of al-Kassani (My own translation take it carefully):

Ibn al-Humam (Hanafi) in his Fath al-Qadir فتح القدير only confirms the sitting or pausing between two sets of taraweeh as a custom of the people of the two harams (Mekka and Medina). So what people have done in between was up to them and that they make tasbeeh etc. is a possible option. He also made clear that people of Medina used to pray during this 4 optional raka'a's.

وأما الاستراحة بعد خمس تسليمات فهل يستحب ؟ قال بعضهم : نعم ، وقال بعضهم : لا يستحب وهو الصحيح ; لأنه خلاف عمل السلف والله الموفق .
There's a dispute whether the pause after the fifth taseleem (tarweehah) is mustahab (recommended)? Some said yes others said it is not recommended and this is the correct view as this is in opposition to what our Sakaf have done and Allah is the bringer of success.

Ibn al-Humam (Hanafi) in his Fath al-Qadir فتح القدير only confirms the sitting or pausing between two sets of taraweeh as a custom of the people of the two harams (Mekka and Medina). So what people have done in between was up to them and that they make tasbeeh etc. is possible option. He also made clear that people of Medina used to pray during this 4 optional raka'a's.