The Blue Posts burned down in 1870, and was replaced by 1878 with The Old Blue Posts (site now redeveloped).
Naming might not be that simple. Portsmouth Encyclopedia has a reference to the Old Blue Posts Tavern in 1865.
But perhaps it was rebuilt twice:
When Peter Simple travelled down to Portsmouth for the first time, to join his ship, he asked the coachman which was the best inn there, and received for reply:
The Blue Postesses
Where the midshipmen leave their chestesses,
Call for tea and toastesses,
And sometimes forget to pay for their breakfastesses.
The “Blue Posts” inn was burned down in 1870, but many who had known it made a renewed acquaintance with the house in 1891, at the Chelsea Naval Exhibition, where a reproduction attracted much attention.
— The Old Inns of Old England, Charles George Harper, 1906
Shades of Gollum? I wonder if Tolkien ever read Peter Simple.
The only references I’ve found to The Blue Posts being rebuilt for the Chelsea Naval Exhibition (Royal Naval Exhibition 1891) are from contemporary newspaper accounts. The official catalogue and guide doesn’t mention anything.
The Keppel’s Head, the George, and The Blue Posts, famous Portsmouth inns patronised by seamen in the time of the press gang, are to be numbered among the historic links of interest.
— Lloyd’s Weekly Newspaper, Sunday 05 April 1891, p5
…while even the refreshment rooms represent famous Portsmouth inns—the Blue Posts, the George, &c.
— The Graphic, Saturday 02 May 1891, p24
The days of Mr. Midshipman Easy […] will be recalled by the signs of the places of refreshment which are named after famous old hostelries at Portsmouth, once the chief places of resort of midshipmen and naval officers. There are “The George,” and the “Blue Posts,” and the “Keppel’s Head,”
— Western Morning News, Monday 30 March 1891, p6
So it looks like they weren’t so much reproductions as themed restaurants. Oh well. Still looking for a photo of the dining rooms. I’ve found one taken from the dining rooms.
Combining the newspaper descriptions with this plan of the site (source), I’m thinking the three “inns” are the dining rooms in the top left corner.
The site is built up of course now, but you can still see the outline of the plot on the ground today: