You do wonder if Francis reinvented himself for some reason, and if he wasn’t really who he said he was, and/or not born when and where he claimed. In 1911, he’s in Woolwich Barracks as a Sergeant, age 38 years, and so another YOB (1872/3), but still born in Plymouth. However, there are only 13 civil birth registrations for Woodhouse in Plymouth and immediate areas from 1863-83 and none look like Francis George.
Francis’ wife, Ellen, was born in Budleigh in Devon (according to the 1911 census) and it may be her in 1881 in Chapel Street, East Budleigh, with parents William (Naval Pensioner) and Elizabeth Britton, although her POB is Morice Town (Devonport). In 1891, she’s still with parents in East Budleigh, and also two Gibbs cousins born in Australia. In 1901, she’s in Plymouth as a domestic born Budleigh.
Interestingly, there is also a Woodhouse family in East Budleigh. In 1881, there's George G Woodhouse, age 50 yrs, born Sedgley in Staffs, clerk in holy orders living at Otterbourne House along with his wife, four children, and various pupils and servants.
I wonder if Francis was born as someone else, taken in by George Woodhouse, and given his name (or taken his name years later). Might be worth digging into this a bit more.
EDIT: In 1881, the Woodhouse children are Mary b 1859 Shropshire, George E b 1867 Budleigh, Lawrence b 1874 Budleigh and Emmeline D b 1876 Budleigh. There are birth registrations for Mary and Emmeline (Dorothea) with mother’s maiden name Benson. Also brs for male Woodhouse Q2 1866, female Woodhouse Q2 1868 and John Woodhouse Q3 1871, all mmn Benson and in St Thomas (the Budleigh registration district).
The male Woodhouse born 1866 could be George E (George Edward bp 1866 East Budleigh), and the female Woodhouse br 1868 could be Ruth Martin Woodhouse (bp 1868 East Budleigh) but I can’t find a br for Lawrence, although he was bp 1873 East Budleigh. He seems to be a pupil in Great Amwell in 1891, and a school teacher in Malvern in 1901, but might there be other children to George and Elizabeth Woodhouse without an obvious birth registration?