The present work is devoted to the development of a method for high-precision tomography of ion qubit registers under conditions of limited distinguishability of the states of a logical value 0 and a logical value 1. In the considered ion qubits, the identification of the quantum state is achieved by measuring the fluorescence of the ion by repeated excitation of the cyclic transition, which includes only the lower energy state that sets a logical value 0 and becomes bright, but does not include the upper metastable state that remains dark and sets logical value 1. It is important to note that it is not always possible to achieve low levels of registration errors due to the finite lifetime of excited levels, photon scattering, dark noise, low numerical aperture values, etc. However, even under such conditions, with use of the model of fuzzy quantum measurements, it is possible to provide precise control of quantum states. We show that a model that is characterized by relatively high levels of errors under conditions, where we have a reliable statistical model of their occurrence, is more accurate than the case when the considered errors are small, but we do not have an adequate statistical model for the occurrence of these errors. In the given illustrative examples, we show that the factor of reducing the loss of accuracy with the use of the model of fuzzy measurements can reach values of the order of 1000 or more in comparison with standard measurements.