An interesting thought that springs to mind about the experience of walking a city but using a map that doesn’t ‘match’ the actual space, is Guy Debord and his writings upon the Situationists International.
A classic example might be Debord’s friend who walk Germany with a map of London…
“The production of psychogeographic maps, or even the introduction of alterations such as more or less arbitrarily transposing maps of two different regions, can contribute to clarifying certain wanderings that express not subordination to randomness but complete insubordination to habitual influences (influences generally categorized as tourism that popular drug as repugnant as sports or buying on credit). A friend recently told me that he had just wandered through the Harz region of Germany while blindly following the directions of a map of London This sort of game is obviously only a mediocre beginning in comparison to the complete construction of architecture and urbanism that will someday be within the power of everyone.”
Guy-Ernest Debord, Introduction to a Critique of Urban Geography