Hello again,
I've looked for every permutation of the names on the 1871 England census, but couldn't find Annie and her parents.
I looked again at the 1881 which you mentioned, - it's very odd - the children William and Annie are both shown as aged 12, and Henry in the middle has his age crossed out! Then I noticed the last two children were born in Wales...
So I looked at the Welsh 1871 census, and found somethone who may or may not be your Annie Greenway....
it's at RG10/5582 Folio 27, Page 6
Brecknock, Breconshire, Wales,
William Greenway, 25, Farm labourer, born Radnorshire, Wales
Elizabeth, 22, wife, born Radnorshire,
Ann daughter, aged 18 months, born Cusop, Breconshire
Ann Prosser, 26, sister in law, domestic servant, born Radnorshire.
I'm pretty sure there is no Cusop in Wales - though it is pretty near to Wales.
I do hope this is the right one, because it'll be handy to have an unmarried sister in law called Ann Prosser living with them - as that would be Elizabeth's maiden name.
And hey presto, on Free BMD, there is a William Greenway marrying an Elizabeth Prosser - December quarter, Hay, 11b 289.
Hope they're the right ones.
Yorkslass