If you have 3 indefinite nouns, then you need to spot the division into units of meaning, as Labhrás has said:
1. [tigh tábhairne] baile: a townland's pub
2. tigh [sagart paróiste]: a parish priest's house
These units of meaning may have lenition within them for reasons that relate to the pattern of lenition on the unit of meaning as a separate phrase:
1. [clann mhac] duine: a man's sons [mhac lenited after the feminine noun clann in the 'inead brí']
2. tigh [bean chaointe]: a wailing woman's house
In all four examples, the genitive is not declined.
If you decline for the genitive (a rarer choice, as the genitive is definitely on the way out in Irish):
tigh pobail bhaile (lenited according to 4.14), tigh sagairt pharóiste (lenited according to 4.14, but I have found the definite "an tsagairt paróiste" in Muskerry literature; Seanchas Chléire has both "an tsagairt paróiste" and "an tsagairt pharóiste", so the unlenited versions may be typos; note that there are examples of "an tsagairt paróiste" on gaois.ie), clann mhac duine, tigh mná caointe.
Now if the governing noun is feminine, there is an argument to lenite after it. But note this sentence from Peadar Ua Laoghaire:
Quote:
"Tá giolla agam," ar seisean, "agus inghean mic inghíne file ab eadh seana-mháthair a mhná."
"I have a manservant, and his wife's grandmother was a daughter of a son of a daughter of a poet".
The link to that passage is
https://archive.org/details/guaire02olea/page/146/mode/2up (right-hand page, p147, line 66)
Ua Laoghaire has "iníon mhic" in a different passage in his works - the issue of whether to lenite after a feminine noun is bedevilled, and there may be a typo in the passage above. [Iníon mhic] as a unit of meaning would call for lenition. Possibly, the fact that iníon mic stands in a longer concatenation of genitives (iníon mic iníne file) may rupture the feeling of iníon mhic as an inead brí and this may explain non-lenition?
Also note this from Peadar Ua Laoghaire's Mo Sgéal Féin:
Quote:
Ar an ngaraidhe prátaí iseadh mhaireadh gach aon duine bocht an uair sin, agus ar pé braon bainne a gheibheadh sé ó'n bhfeirmeóir go mbíodh slígh fir oibre aige uaidh
Slí is feminine, but does not lenite fir oibre (a declined genitive). ([Slí bheatha] and [slí mhaireachtaint] do have lenition, because there is an inead brí.)
For this reason, I
feel that "oifig [pobail bhaile]" is right, without lenition, at least where the genitive is declined. Where the genitive is not declined, I think you need lenition to show the relation between "pobal baile" and "oifig": oifig [phobal baile]. The discussion in Graiméar Gaeilge 4.15 and 4.16 simply doesn't clarify either way.
But if the mental division into phrases is as follows, then you need leniton: [oifig phobail] baile. (I don't see the need for lenition of the b, because it is not in the inead brí.)
The passage in Graiméar Gaeilge on this is extraordinarily convoluted, and so I have not been able to work out if my understanding is correct or not.