The highest efficiency of wave energy capture by point absorbers is attained under conditions close to resonance. In most cases, the natural frequency of resonance is higher than the typical frequency of the waves. Reactive phase control has been proposed to improve these situations. Reactive power contributes nothing to the average delivered power and is back-and-forth exchange of energy between the power take-off system (PTO) and the oscillating system. Apart from larger peaks in PTO forces and power, a major drawback of reactive phase control is the energy loss by dissipative processes inherent to the back-and-forth energy exchange, especially if the magnitude of such exchanged energy is comparable to, or even significantly larger than, the net absorbed energy, which may be the case in point absorbers. The paper considers a floating axisymmetric two-body wave energy converter constrained to oscillate in heave in deep water. The outer body is a buoy, whereas the inner body is a long surface-piercing cylinder against whose inertia the outer floater reacts. The PTO converts the relative motion between the two bodies and the associated forces into useful energy. The basic equations in the frequency domain for the reactively controlled two-body converter performance in regular waves are presented for an imperfectly efficient PTO. If the inner cylindrical body is fixed, the equations are reduced to a single-degree-of-freedom. The analysis is extended to irregular waves characterized by a variance density spectrum. Numerical results are presented for regular and irregular waves, including reactive control optimization.