  We prove higher Sobolev regularity for bounded weak solutions to a class of
nonlinear nonlocal integro-differential equations. The leading operator
exhibits nonuniform growth, switching between two different fractional elliptic
``phases" that are determined by the zero set of a modulating coefficient.
Solutions are shown to improve both in integrability and differentiability.
These results apply to operators with rough kernels and modulating
coefficients. To obtain these results we adapt a particular fractional version
of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work
``Nonlocal self-improving properties" Anal. PDE, 8(1):57--114 for the specific
nonlinear setting under investigation in this manuscript.
