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Quantum key distribution (QKD) provides information-theoretic security based on the laws of quantum mechanics. The desire to reduce costs and increase robustness in real-world applications has motivated the study of coexistence between QKD and intense classical data traffic in a single fiber. Previous works on coexistence in metropolitan areas have used wavelength-division multiplexing, however, coexistence in backbone fiber networks remains a great experimental challenge, as Tbps data of up to 20 dBm optical power is transferred, and much more noise is generated for QKD. Here we present for the first time, to the best of our knowledge, the integration of QKD with a commercial backbone network of 3.6 Tbps classical data at 21 dBm launch power over 66 km fiber. With 20 GHz pass-band filtering and large effective core area fibers, real-time secure key rates can reach 4.5 kbps and 5.1 kbps for co-propagation and counter-propagation at the maximum launch power, respectively. This demonstrates feasibility and represents an important step towards building a quantum network that coexists with the current backbone fiber infrastructure of classical communications., Comment: 11 pages, 6 figures. Accepted for publication in Optics Express