This paper describes a 3D implementation of the sharp-interface theory for material heterogeneities and is, hence, able to identify equilibrium shapes of precipitates in superalloys. The theory is adopted from Morton E. Gurtin and extended by crystal plasticity in the bulk. Crystal plasticity relaxes stresses at the phase interface, which leads to subsequent coalescence of the precipitates. The fully implicit model employs the extended finite element method (XFEM) in conjunction with level sets. The level set is advected in a velocity field computed by the stress-modified Gibbs-Thomson interface condition. Mechanical equilibrium and level set update are solved in a staggered procedure. Jump quantities are treated by means of a suitable enriched least square smoothing. Multiple schemes for the computation of curvature of surfaces in the context of the XFEM are presented and compared. Equilibrium shapes at different levels of misfit are computed. A cuboidal equilibrium shape is retrieved in a rotated mesh in order to quantify mesh-independence, a linear volume-time relationship during Ostwald ripening is reproduced and merging of particles under tension is reported.