Which of these two sentences are correct?
a) My other sister is taller than me
b) My another sister is taller than me
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Each noun phrase has one determiner slot. It can't have two determiners in a row.
[My other sister] is taller than me.
This sentence is fine.
In the noun phrase my other sister, the determiner slot is filled by my, a genitive pronoun. The following word other is an adjective, not a determiner.
[Another sister] is taller than me, too.
This sentence is also fine.
In the noun phrase another sister, the determiner slot is filled by another, a determinative. This word was formed by compounding an + other.
I'm sure you already know you can't put my and an together, because that would be two determiners together. For the same reason, you can't put my and another together.
*[My another sister] is taller than me.
This sentence is ungrammatical.
Why? Because the noun phrase only has one determiner slot, but you've tried to fill it twice.
In this answer, the * symbol marks a sentence as ungrammatical.
(Reference: The Cambridge Grammar of the English Language (2002), p.391)