  We propose and demonstrate the appearance of an effective attractive
three-body interaction in coherently-driven two-component Bose Einstein
condensates. It originates from the spinor degree of freedom that is affected
by a two-body mean-field shift of the driven transition frequency. Importantly,
its strength can be controlled with the Rabi-coupling strength and it does not
come with additional losses. In the experiment, the three-body interactions are
adjusted to play a predominant role in the equation of state of a cigar-shaped
trapped condensate. This is confirmed though two striking observations: a
downshift of the radial breathing mode frequency and the radial collapses for
positive values of the dressed-state scattering length.
