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Andrew, that's what I was hoping to hear, it's preferable for it to be two different babies. Is just odd that I can find no furthur trace of the 'wrong' Jemima. I am inclined to think that the wrong one is still related to me through Frances, Joseph's sister.
Merry Christmas.

Well, the cert has come already, and now I'm even more confused than before. So the cert I already had was numbered 21, 5th May 1885, no address given, just Wingate R. S. D, Jemima Smith, mother:Frances Ann Smith, no father named, registered on 2nd June 1885, informant F. A. Smith, mother.
This new one is numbered 20, born 5th May 1885, Wingate R. S. D, Jemima Smith, mother:Eliza Smith formerly Morris, father Joseph Smith, coal miner, dated 2nd June 1885. Same registrar on both.
What on earth can I deduce from this??
Also of note is that I share 115cM of DNA with a cousin related through Eliza Morris's brother, and my mum 188cM, just to give you all the facts.
Thoughts welcomed 😁

So I'm not the only one this has happened to!
I have actually just double checked my records and think I may have solved part of the muddle. I don't think my Jemima was born on May 5th after all, having looked through a copy of the family bible transcriptions I now think that I must have added in 5th May 1885 to my tree after getting the wrong cert as her birth date isn't recorded in it as I had imagined it was. So I'm hoping that when I get the right one (they say on 31st Dec) that I will have a new birth date for my Jemima. If that is the case then it'll definitely be two separate babies, the puzzling things will be trying to ascertain whether the mother of this second one was Frances Ann Smith, who was the sister of Joseph Smith, father of my Jemima. That seems likely to me as Joseph and Frances's deceased mother was called Jemima. Also, what on earth happened to the second Jemima, or Mary Jemima from her christening record as I can't find any more evidence of her existence.

Now that baptism would seem to indicate it could be two babies born on the same day! Thank you for finding that. Still confusing though as I can't seem to find any sign of this Mary Jemima on any censuses, nor on deaths. Admittedly I'm only looking at Wingate (Easington) as with Smith as a surname it's a bit difficult to look more widely.

Andrew,
I'm tending to think you may be correct, it does seem awfully strange, and I've just had a very quick look on the 1891 census and can only see one Jemima Smith that fits (need to check deaths now, just in case). It's just weird that I have the Morris DNA but Eliza did have brothers so that would perhaps be a possibility as to who the father could have been, although I'm never really going to know.