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We present a theory of tunneling spectroscopy of spin-selective Aharonov-Bohm oscillations in a lateral triple quantum dot molecule. The theory combines exact treatment of an isolated many-body system with the rate equation approach when the quantum dot molecule is weakly connected to the leads subject to arbitrary source-drain bias. The tunneling spectroscopy of the many-body complex is analyzed using the spectral functions of the system and applied to holes in a quantum dot molecule. Negative differential conductance is predicted and explained as a result of the redistribution of the spectral weight between transport channels. It is shown that different interference effects on singlet and triplet hole states in a magnetic field lead to spin-selective Aharonov-Bohm oscillations.