Laighneach wrote: "
An t-aran a rinn mo sheanmhair an-dè"/"the bread that my grandmother made yesterday" is
not a noun phrase?
"a rinn mo sheanmhair an-dè" is an adjectival clause; the whole can be replaced with a possessive adjective: "
tha mi ga ithe".
Perhaps I've mis-used the terminology, grammar isn't my strongpoint
Anyway, I take it that it would be in the genitive regardless of how the definite noun is qualified,
and that the likes of the following would be in the genitive too? :-
Bha i a' biadhadh an eich aca.
Tha e a' glanadh càir Sheumais.
Tapadh leibh a-rithist
OK, so...
"it would be in the genitive regardless of how the definite noun is qualified"
The article only attaches to the last noun in the "string", as you clearly already know from the examples you give -- there is no article beside "càr" because it is not the last* noun in the string. Now it just so happens that the exact same rule applies for the genitive -- you cannot place a genitive where you couldn't place an article. (So e.g. a' glanadh càr Sheumais -> a' glanadh càr
an fhir).
(* Perhaps "deepest" would be a better term than "last", but that's more detail than we need at the moment.)
Now, if you look at your original sentence, you'll see that you have an article before bread, and a possessive "mo" before granny. That's absolutely correct. And because you can correctly put the article before bread, you can correctly put bread in the genitive.
So you actually know all this.
Now, if you want a more technical explanation, "a rinn mo sheanmhair an-dè" is part of the noun phrase, but it is not itself a noun phrase. It is a "relative clause" which is technically an adverbial (i.e. the same class as "to the moon", "over there", "yesterday" etc). Yes, there is a noun phrase within the adverbial ("mo sheanmhair") but "a rinn mo sheanmhair an-dè" is not itself a noun phrase. So "an arain" is in the genitive because it is not followed by a noun phrase.