This lateral poser struck me as I was travelling through Fulham Broadway
today. Took a few minutes to work out, and it was quite satisfying when the
solution hit me:

Until quite recently, the only entrance/exit to/from Fulham Broadway station
was off the southern end of the platforms.

After the new booking hall opened, the old stairs were blocked off, and now
the only exit is to the northern end of the platforms - e.g. the way out has
moved from one end of the station to the other.

However, the station has 'way out' signs dating from the 1950s or earlier...
which point the correct way to the new exit.

BTN