  We study the number of distinct ways in which a smooth projective surface $X$
can be realized as a smooth toroidal compactification of a ball quotient. It
follows from work of Hirzebruch that there are infinitely many distinct ball
quotients with birational smooth toroidal compactifications. We take this to
its natural extreme by constructing arbitrarily large families of distinct ball
quotients with biholomorphic smooth toroidal compactifications.
