A tissue of lies
A story invented in order to deceive.
If you use a search engine to search for 'tissues' you will find many pictures of paper handkerchiefs and of human skin - that's what 'tissue' means to us these days. So, whence the phrase 'a tissue of lies'? It might be thought that the meaning derives from the filmy nature of tissue and that this had been taken up as a metaphor for lies that were easy to see through and would readily break down on examination. That's not an unreasonable assumption but is in fact completely wide of the mark.
For the correct meaning of 'tissue of lies' we have to go back to the 14th century meaning of 'tissue', that is, 'an intricately woven ornamental cloth'. That meaning is first recorded in the Middle English allegorical poem The Romaunt of the Rose, circa 1366:
The barres [decorative straps] were of gold ful fyne, Upon a tyssu of satyne.
The defining characteristic of tissues was the complexity of their weaving. A 'tissue of lies' is a complex, interwoven series of lies, not a flimsy and unconvincing one.
The figurative meaning of the intermingling of characteristics, usually of a bad kind, began to be used in the 18th century. From then onward, any combination of 'a network/web/fabric/tissue of absurdity/error/falsehood' can be found somewhere in literature. Of these, only 'tissue of lies' has withstood the test of time. The phrase began to be used in the early 19th century, as in this example from the London journal, The Monthly Review, January 1800:
The ingenuity and cunning of politicians are not infrequently employed to conceal or misinterpret facts; and venal writers are easily found, ready to construct a tissue of lies to serve the purposes of their employers.
By the way, there's no truth that the word 'atishoo' derives from the handkerchief meaning of 'tissue' - that's just atishoo of lies.