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Quantum phase is not directly observable and is usually determined by interferometric methods. We present a method to map complete electron wave functions, including internal quantum phase information, from measured single-state probability densities. We harness the mathematical discovery of drum-like manifolds bearing different shapes but identical resonances, and construct quantum isospectral nanostructures with matching electronic structure but divergent physical structure. Quantum measurement (scanning tunneling microscopy) of these “quantum drums”—degenerate two-dimensional electron states on the copper(111) surface confined by individually positioned carbon monoxide molecules—reveals that isospectrality provides an extra topological degree of freedom enabling robust quantum state transplantation and phase extraction.