A tunable and balanced heat interferometer is proposed and analyzed. The device consists of two superconductors linked together to form a double-loop interrupted by three Josephson junctions coupled in parallel. Both superconductors are held at different temperatures allowing the heat currents flowing through the structure to interfere. As we show here, thermal transport is coherently modulated through the application of a magnetic flux. Furthermore, such modulation can be tailored at will through the application of an extra control flux. In addition we show that, provided a proper choice of the system parameters, a fully balanced interferometer is obtained. The latter means that the phase-coherent part of heat current can be controlled to the extent of being fully suppressed. Such a device allows for a versatile operation appearing, therefore, as an attractive key to the onset of low-temperature coherent caloritronic circuits.