The phrase isn't about actual tissues but refers metaphorically to a complex network of lies. The court saw through his testimony, which was nothing but a tissue of lies.
Pro Tip 2/3
Negative Connotation
Using 'a tissue of lies' implies a strong disapproval and condemnation of the dishonesty. The journalist exposed the politician's promises as a tissue of lies.
Pro Tip 3/3
Emotional Impact
The phrase often evokes feelings of betrayal and disappointment in the listener. Finding out her hero's memoir was a tissue of lies left her feeling deeply disillusioned.