  We give a complete description of the possible ranges of real Smirnov
functions (quotients of two bounded analytic functions on the open unit disk
where the denominator is outer and such that the radial boundary values are
real almost everywhere on the unit circle). Our techniques use the theory of
unbounded symmetric Toeplitz operators, some general theory of unbounded
symmetric operators, classical Hardy spaces, and an application of the
uniformization theorem. In addition, we completely characterize the possible
valences for these real Smirnov functions when the valence is finite. To do so
we construct Riemann surfaces we call disk trees by welding together copies of
the unit disk and its complement in the Riemann sphere. We also make use of
certain trees we call valence trees that mirror the structure of disk trees.
