I've looked up the etymology of tante, but it just says it comes from Old French ante without saying how it gained the initial "t".
Where did this "t" come from?
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The Littré has this hypothesis, ta ante has agglutinated in tante which would have turn into a noun.
It gives some examples in Walloon with which monfré is brother and mononk is uncle.
I found: kimin s' poitt voss monfrér ? = Comment se porte votre frère ?
A similar agglutination exists with monsieur where mon has lost its adjectival status: ton monsieur, mon bon monsieur are possible.