I'm going to throw this in just as a speculative punt, until it can be ruled in or out, the detail can be found in NLI RC parish records and in civil records for the 1864+ births, I'll be as brief as I can, in case it's a red herring.
It's a couple called John O'Brien and Bridget Egan who married in 1855 in Ss Peter & Paul's Clonmel, Tipperary, within the Diocese of Waterford and Lismore, which covers both counties Tipperary and Waterford. In the following, sometimes the surname O'Brien is used, other times it's just Brien.
I can see a run of children for this couple, all baptised in Ss Peter & Paul's Clonmel, Tipperary, except one. Those baptised in Ss Peter & Paul's Clonmel are Thomas John Brien (1858), Anastasia Brien (1860), Ambrose O'Brien (1862), Bridget O'Brien (1865), John Brien / O'Brien (1867), Frederick O'Brien (1869) and Patrick Brien / O'Brien (1872). The baptismal records don't mention any address for these folk, the 1864+ civil birth records give their address as different streets in Clonmel, Tipperary.
Between Frederick and Patrick, there is a baptismal record for an Anne Brien who was baptised in Castletown Geoghan RC parish, County Westmeath, it took place on 19 November 1870, parents recorded as John Brien (soldier) + Bridget Egan, the sponsor was Anne Daly (right page):
https://registers.nli.ie/registers/vtls000635098#page/15/mode/1upWhen I went looking for a corresponding civil birth for Anne, the only record I could find shows her mother's maiden name as Owens - I'm not sure why, though I'm aware of sources that say Egan is ultimately descended from Eoghan, for which Owen is the anglicised version. Anyhow, the civil birth was registered in Castletown Geoghan, County Westmeath, Anne was born on 15 November 1870 to 'John Brien 89th Regiment India' and Bridget Owens, I can't make out the address, I thought it was maybe the townland of Garhy, but it looks like there's a 't' in there, so maybe not:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1870/03321/2217113.pdf As reference, here are the civil birth registrations for Frederick and Patrick who were born before and after Anne - I'm wondering if Bridget maybe travelled from Clonmel to whatever that place is in County Westmeath to vist friends / relatives and have her child, while John was away soldiering.
Frederick in 1869:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1869/03381/2239592.pdfPatrick in 1872:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1872/03258/2193917.pdfAnyhow, see what you think, it does tick some boxes, for example the birth day and month is close and the geography of Clonmel in Tipperary and Castletown Geoghan in Westmeath sort of covers all bases, in terms of what you've mentioned as her birthplace in the various censuses.