  In this work we consider an inhomogeneous two-phase obstacle-type problem
driven by the fractional Laplacian. In particular, making use of the
Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas
and blow-up analysis, we provide a classification of the possible vanishing
orders, which implies the strong unique continuation property. Moreover, we
prove a stratification result for the nodal set, together with estimates on its
Hausdorff dimensions, for both the regular and the singular part. The main
tools come from geometric measure theory and amount to Whitney's Extension
Theorem and Federer's Reduction Principle.
