Inspired by a discussion to an answer on this question, has a movie which was explicitly adapted from a novel (regardless of how faithful the adaptation was) received an official novelisation based solely, or even primarily, on the movie?
I think the answer to this is probably many
After a short think, the first example I was able to find was:
Do Androids Dream of Electric Sheep?, by Philip K. Dick was adapted as
Blade Runner, directed by Ridley Scott, which in turn was adapted as
Blade Runner: A Story of the Future by Les Martin
Philip K. Dick approved of the movie despite the differences in the adaptation, but refused to write the novelization and is quoted as saying
[I was] told the cheapo novelization would have to appeal to the twelve-year-old audience.
Planet Of The Apes (1968) was based on French author Pierre Boulle's 1963 novel La Planète des Singes. After becoming a movie and additional sequels, it was further developed into cartoons, TV series and comic books.
Additionally, all of the original sequels spawned novelizations by established science fiction writers of the day, each of which went through multiple reprintings of their own.
- Michael Avallone wrote the novelization for Beneath the Planet of the Apes in 1970.
- Jerry Pournelle, who later co-authored Lucifer's Hammer and The Mote in God's Eye, wrote the Escape from the Planet of the Apes novelization.
- John Jakes, former Science Fiction Writers of America president, wrote Conquest of the Planet of the Apes.
- David Gerrold, scriptwriter for the Star Trek episode "The Trouble with Tribbles", novelized Battle for the Planet of the Apes.
Novelizations of the live action and animated television series were also produced. William T. Quick novelized the 2001 Planet of the Apes; he also wrote two prequel novels, and several other book tie-ins were published.
Source: Planet Of The Apes Wiki
If you consider a graphic novel to be a novel, then the graphic novel V for Vendetta (written by Alan Moore and illustrated by David Lloyd) was adapted as a film (directed by James McTeigue and written by the Wachowskis), which was in turn novelized by Steve Moore (no relation to Alan Moore).
The film had a number of differences from the graphic novel, and these differences are carried through to the novelization.
In 1992 Francis Ford Coppola did his version of Dracula, released under the title "Bram Stoker's Dracula". Apparently the original novel was not adequate so a novelization was published (by James V. Hart and Fred Saberhagen).
Another example, years ago someone showed me a copy of H. Rider Haggard's "King Solomon's Mines" with a picture of Stewart Granger and Deborah Kerr from the 1950 movie on the cover. I know Haggard's writing; I read several of his books. He showed me the first page - it certainly was not Haggard. The cover and spine say Haggard. On the title page at the bottom it gives the movie director credit, the producer credit, screenplay credit and "Fictionized by Jean Francis Webb". I found the book on eBay so I could report this.
One relatively clear-cut example is The Wicker Man, which "is a 1978 horror novel written by Robin Hardy and Anthony Shaffer. It was based on the 1973 cult horror film The Wicker Man, directed by Hardy and written by Shaffer. [..] The film itself is loosely based on the 1967 novel Ritual by the actor and novelist David Pinner."
If you'll expand your question to accept "Novels based on movies starring characters who originated in print" then The 2011 Conan the Barbarian Novel is 'Based on the Movie', and there are several James Bond novels based on the movies, and lots of the Marvel universe movies have been subsequently novelised.
An early almost-example is the novel The Circular Staircase, which was adapted into the play The Bat, which was separately adapted into novel The Bat and movie The Bat.
As no-one seems to have mentioned them yet, James Bond, The Spy Who Loved Me and James Bond and Moonraker are novelisations of the relevant Bond films, which have no resemblance to the original novels by Ian Fleming whose titles they bear.
(well, the OP did say "regardless of how faithful the adaptation was")
Another example to the great ones given previously:
In 1937, Frederic Dannay and Manfred Lee (writing as Ellery Queen) released The Door Between.
In 1940, it was adapted as Ellery Queen, Master Detective, directed by Kurt Neumann and written by Eric Taylor.
Finally, in 1941 the novelization of the movie was published as Ellery Queen Master Detective (sometimes reissued as Vanishing Corpse). The ghostwriter was Laurence Dwight Smith, but the cover credited Ellery Queen only.
Hence, this is an example where both the source and the novelization were published with the same name of the author on the cover.