Looking at the 1901 census, John Mulligan was 50 so born c 1850. That’s well before the start of statutory birth registration in Ireland (Vital Records). They start in 1864. So you would need to search for his baptism in the local church records. There are 2 likely Presbyterian churches in the Clogher area. Aughentaine has baptism from 1836 onwards. Clogher or Carntall has baptisms for 1819 to 1845 and then 1858 onwards. So there’s a gap around 1850. The records are not on-line so far as I am aware but there are copies in PRONI (the public record office) in Belfast.
I searched the statutory records for Mary Mulligan born in the Clogher area in 1877. There wasn’t one. The 3 nearest were: 1870, 1879 & 1880. You can view them on this site: https://churchrecords.irishgenealogy.ie/churchrecords/
Irish people in the 1800s didn’t celebrate birthdays and frequently had little idea of when they were born. If officialdom asked them for an age or date of birth they often just made one up. So I wouldn’t rely too heavily on April 1877 as being Mary’s date of birth. But if it is at all accurate it means John Mulligan was 27 years older than her. Therefore they are most unlikely to be brother and sister.
