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Stabilizer codes allow for non-local encoding and processing of quantum information. Deformations of stabilizer surface codes introduce new and non-trivial geometry, in particular leading to emergence of long sought after objects known as projective Ising non-Abelian anyons. Braiding of such anyons is a key ingredient of topological quantum computation. We suggest a simple and systematic approach to construct effective unitary protocols for braiding, manipulation and readout of non-Abelian anyons and preparation of their entangled states. We generalize the surface code to a more generic graph with vertices of degree 2, 3 and 4. Our approach is based on the mapping of the stabilizer code defined on such a graph onto a model of Majorana fermions charged with respect to two emergent gauge fields. One gauge field is akin to the physical magnetic field. The other one is responsible for emergence of the non-Abelian anyonic statistics and has a purely geometric origin. This field arises from assigning certain rules of orientation on the graph known as the Kasteleyn orientation in the statistical theory of dimer coverings. Each 3-degree vertex on the graph carries the flux of this "Kasteleyn" field and hosts a non-Abelian anyon. In our approach all the experimentally relevant operators are unambiguously fixed by locality, unitarity and gauge invariance. We illustrate the power of our method by making specific prescriptions for experiments verifying the non-Abelian statistics.