  We prove a structure theorem for the solutions of nonlinear thin two-membrane
problems in dimension two. Using the theory of quasi-conformal maps, we show
that the difference of the sheets is topologically equivalent to a solution of
the linear thin obstacle problem, thus inheriting its free boundary structure.
More precisely, we show that even in the nonlinear case the branching points
can only occur in finite number. We apply our methods to one-phase free
boundaries approaching a fixed analytic boundary and to the solutions of a
one-sided two-phase Bernoulli problem.
