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We consider abstraction-based design of output-feedback controllers for dynamical systems with a finite set of inputs and outputs against specifications in linear-time temporal logic. The usual procedure for abstraction-based controller design (ABCD) first constructs a finite-state abstraction of the underlying dynamical system, and second, uses reactive synthesis techniques to compute an abstract state-feedback controller on the abstraction. In this context, our contribution is two-fold: (I) we define a suitable relation between the original system and its abstraction which characterizes the soundness and completeness conditions for an abstract state-feedback controller to be refined to a concrete output-feedback controller for the original system, and (II) we provide an algorithm to compute a sound finite-state abstraction fulfilling this relation. Our relation generalizes feedback-refinement relations from ABCD with state-feedback. Our algorithm for constructing sound finite-state abstractions is inspired by the simultaneous reachability and bisimulation minimization algorithm of Lee and Yannakakis. We lift their idea to the computation of an observation-equivalent system and show how sound abstractions can be obtained by stopping this algorithm at any point. Additionally, our new algorithm produces a realization of the topological closure of the input/output behavior of the original system if it is finite-state realizable.