T. S. Eliot has a reputation as a difficult and serious poet. He also wrote very serious essays about other serious authors. However, according to Johannes Kleinstück (T. S. Eliot - mit Selbstzeugnissen und Bilddokumenten. Reinbek bei Hamburg: Rowohlt, 1966) he was also a practical joker who bought exploding cigars and cushions that made funny noises when you sat down on them. (After reading that, it becomes easier to understand how Eliot could later come up with something like Old Possum's Book of Practical Cats.)
Kleinstück even claims that he wrote crime novels ("Kriminalromane") that were published under a pseudonym (page 22):
Angeblich hat er auch unter einem Pseudonym Kriminalromane geschrieben und in seiner Tätigkeit als Verlagsdirektor bei Faber & Faber eine große Zahl von "Waschzetteln" ("blurbs"), deren Autorschaft wohl nie geklärt werden kann.
Apparently he also wrote crime novels under a pseudonym, and, as director at Faber & Faber, a large number of blurbs whose authorship will probably never be resolved.
Unfortunately, Kleinstück doesn't provide a source for this claim; on the internet, I found that Eliot wrote five rules for "detective conduct" but nothing about novels he is supposed to have written.
So what is the source for the claim that T. S. Eliot wrote crime novels? Since Kleinstück's book was published in the year following Eliot's death, the source should be someone who had known him.