I haven't gained ANY traction at all on Joseph Agnew and his wife, but I'll post what I've found regardless, in case it offers up something for fresh eyes to help you.
A transcript of the 1886 church marriage record shows that it took place in St. Peter's, Belfast. I couldn't find a corresponding civil registration record, but it wouldn't be the first time I've encountered a St. Peter's marriage that didn't make it into the civil registration records.
On 18 July 1886, Patrick Murphy of 108 Milford St. married Ellen Agnew of 6 Townsend St., Patrick's parents were recorded as Patrick and Mary Murphy, Ellen's parents as Joseph and Ellen Agnew. The witnesses were recorded as Joseph McSherry and Elizabeth Souters. A comment on the transcript says that there was a dispensation in banns, that Patrick Murphy's parents were dead, and that the bride's mother's name and the witness name 'Elizabeth Souters' were unclear.
I then saw the following births for the couple:
- Mary (b. 1887 at 108 Milford Street, father Patrick a labourer):
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1887/02552/1944302.pdf(a St. Peter's baptismal transcript for Mary identified her sponsors as James Magowan and Anne J Dornan.)
- Rose Ann (b.1889 at 108 Milford Street, father Patrick a labourer):
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1889/02456/1912688.pdf(a St. Peter's baptismal transcript for 'Rosanne' identified her sponsor as Mary Ann Huston.)
Nancy (b. 1891 at 108 Milford Street, father Patrick a watchman):
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1891/02381/1888564.pdf(a St. Peter's baptismal transcript for 'Ann' [ie. Nancy] identified her sponsor as Catherine Quinn.)
Patrick (b. 1892 at 108 Milford Street, father Patrick a pensioner):
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1892/02328/1871730.pdf(a St. Peter's baptismal transcript for 'Patrick John' identified his sponsor as Maryanne Morgan.)
The next thing I saw was this civil death registration for the above Patrick John Murphy in 1895, in which the informant was recorded as
'Ellen Agnew Grand-mother Present at death 108 Milford St'.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1895/05917/4680043.pdfAnd that's pretty well it, in terms of what I found.
The only other thing I would add is that in the Belfast street directories that are available online via the Lennon Wylie and PRONI websites, Patrick Murphy appears at 108 Milford Street from 1884 to 1899, described as a 'pensioner' - which maybe implies an army or RIC background.
I even wondered if this below might be Ellen (Agnew) Murphy and family in the 1911 census, but the problem of course is that all the forenames and the surname in this line are just so thick on the ground.
http://www.census.nationalarchives.ie/pages/1911/Antrim/Smithfield_Ward/Campbell_s_Row/154926/;Anyhow, that's all I could find so far, over to fresh eyes.