Derivative-free optimization (DFO) has enjoyed renewed interest over the past years and especially model-based trust-region methods have been shown to perform well on these problems. We want to present a new interpolation-based trust-region algorithm which can handle nonlinear and nonconvex optimization problems involving equality constraints and simple bounds on the variables. The equality constraints are handled by a trust-region SQP approach, where each SQP step is decomposed into a normal and a tangential step to account for feasibility as well as for optimality. Special care must be taken in case an iterate is infeasible with respect to the models of the derivative-free constraints. Globalization is handled by using an Augmented Lagrangian penalty function as the merit function. Furthermore, our new algorithm uses features of the algorithm BCDFO, proposed by Gratton et. al. (2011), which handles bound constraints by an active-set method and has shown to be very competitive for bound-constrained problems. It relies also on the technique of self-correcting geometry, proposed by Scheinberg and Toint (2010), to maintain poisedness of the interpolation set. The objective and constraint functions are approximated by polynomials of varying degree (linear or quadratic). We present numerical results on a test set of equality-constrained problems from the CUTEr problem collection.