Hi everybody,
Many thanks for the several fascinating comments, which look as if they may give us an answer to the Greetham name.
I had found the list of children of Capt Richard Tyldesley and Ann Stanley, including our own Ann Tyldesley, but I had not done a lot of research on the life of her siblings. It does seem to be an amazing coincidence, so I suspect it is heading in the right direction.

What is curious about it is that to the best of my belief, the Greetham name does not appear in our family until Christiana's son of 1900.

Ann Tyldesley was born in 1737 and married in 1766 (date of her death uncertain as the Malew burials give two possible candidates), but by 1830.

Her son Henry Corrin (b 1769) married Judith Kinvig, and we can trace 3 children but no Greethams.

The grandson, Henry Corrin of 1803 m Charlotte Chapman in 1830 , with 6 issue and no Greetham.

The great grandson, Henry William Corrin of 1833 had seven issue, no Greetham

The great great grand daughter, Christiana Frances Corrin has a son in 1900, Alfred Greetham. It is a surprisingly LONG time for a name of an in-law to remain dormant and then be activated.

On the other hand, with the exception of Judith Kinvig and Charlote Chapman, the two mystery girls, there are no other possible close Greetham associations, and from the number of legends handed down in the family,
tradition clearly did matter.

To cite one example, my mother repeated a firm legend that one branch of the family came from Galloway in Scotland. It is clear that we were descended from Jeffrey Gallowey (who died in 1610) and his granddaughter Ellin Galloway/Stevenson who died in 1704. and in 300 years, it would be very easy for "the galloway family" to become "the family from Galloway". (It is quite possible that it was actually the Galloway family from Galloway)

If that story could linger, then the Greetham connection is perfectly believable.

Now to throw in a possible red herring - My Mother did not know where the Greetham connection came from, but believed that

(a) her mother knew someone called Alfred Greetham
(b) There was possibly some Spanish connection to "Greetham"
(c) Finally, and this may be a total gold plated red herring. In the 1920s, my mother was invited to visit an old freind of her mother's family, a Dr Phillips who was in his 80s or 90s, by 1922, and lived at Claydon, Bucks. The doctor was either a widower or a batchelor, as My motehr coul not recall a wife. He was still in practice, and had a car, I believe a Lanchester, which was driven by a chauffeur who had been the coachman before 1900. He did not approve of driving after dark.

(e) Christiana Frances was SENT to a Ladies Finishing School at Southport. She was born 1863, so that would be 1879-1881 or thereabouts

As to what relevance any of this has to Greetham I really do not know.

Robert