Double-spin asymmetries in exclusive electroproduction of real photons from a transversely polarized hydrogen target are measured with respect to the product of target polarization with beam helicity and beam charge, and with respect to the product of target polarization with beam helicity alone. The asymmetries arise from the deeply virtual Compton scattering process and its interference with the Bethe--Heitler process. They are related to the real part of the same combination of Compton form factors as that determining the previously published transverse target single-spin asymmetries through the imaginary part. The results for the double-spin asymmetries are found to be compatible with zero within the uncertainties of the measurement, and are not incompatible with the predictions of the only available GPD-based calculation.
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