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The addition of a weak oscillating field modifying strongly dressed spins enhances and enriches the system quantum dynamics. Through low-order harmonic mixing the bichromatic driving generates additional rectified static field acting on the spin system. The secondary field allows for a fine tuning of the atomic response and produces effects not accessible with a single dressing field, such as a spatial triaxial anisotropy of the spin coupling constants and acceleration of the spin dynamics. This tuning-dressed configuration introduces an extra handle for the system full engineering for quantum control applications. Tuning amplitude, harmonic content, spatial orientation and phase relation are control parameters. A theoretical analysis, based on perturbative approach, is experimentally validated by applying a bichromatic radiofrequency field to an optically pumped Cs atomic vapour. We measure the resonance shifts produced by tuning fields up to the third harmonic.