In my opinion the OP can use either true story or tale in his headline.
An example of how tale can be used effectively in a title is A Tale of Two Cities by Charles Dickens. It is set in London and Paris before and during the French Revolution, it has a story-like edge to it and we expect to read a recount with a beginning, a middle and an end. In Dicken's novel, the word tale has an epic flavour to it. If I were to read the following title "A Survivor's Tale", I would imagine the description of a person's long and arduous journey through life. An existence of strife and struggle punctuated with episodes of great despair but ultimately, a story of great resilience and hope.
In the end, it depends on the story being told and how it is told. If it relates a series of factual events as in a documentary, then I would choose the expression true story. If the recount is told as if it were a novel, then I wouldn't have any objections with the term tale.
Because many answers are based only on the term tale without considering the OP's original headline, I defy anyone who reads the following to believe it is a short story about fairies, elves, goblins, or fishermen adventures.
A SOVIET LABOR CAMP SURVIVOR’S TALE
The following are all examples containing the term tale for non-fictional narrative purposes.
An emaciated man survived 16 months adrift at sea, eating turtles,
birds and fish and drinking turtle blood, having floated up to 8,000
miles from Mexico to a remote Pacific atoll, it was claimed on Friday.
A Norwegian researcher said that the man – who only speaks Spanish and
has a long beard – was not in a good condition after his 24-foot
fibreglass boat washed up on the reef at Ebon Atoll.
If confirmed, the feat would be an unbelievable tale of survival,
reminiscent of the Tom Hanks film Cast Away or perhaps Ang Lee's
adaptation of Life of Pi.
- Poon Lim was a Chinese sailor man who survived 133 days alone in the South Atlantic after the British merchant ship he was serving in was sunk by German marines on November 23, 1942
King George VI bestowed a British Empire Medal (BEM) on him, and the
Royal Navy incorporated his tale into manuals of survival techniques.
[...] The writer Alfred
Bester later stated that Poon Lim's ordeal was used in his novel The
Stars My Destination, which opens with a man stranded in space.
And for fictional purposes
Thirteen-year-old Brian Robeson learned to survive alone in the
Canadian wilderness, armed only with his hatchet. Finally, as millions
of readers know, he was rescued at the end of the summer. But what if
Brian hadn’t been rescued? What if he had been left to face his
deadliest enemy—winter? Author, Gary Paulsen, three-time Newbery Honor
winner, raises the stakes for survival in this riveting and inspiring
story as one boy confronts the ultimate test and the ultimate