We investigate the phase structure of a four dimensional SO(4) invariant lattice Higgs-Yukawa model comprising four reduced staggered fermions interacting with a real scalar field. The fermions belong to the fundamental representation of the symmetry group while the three scalar field components transform in the self-dual representation of SO(4). We explore the phase diagram and find evidence of a continuous transition between a phase where the fermions are massless to one where the fermions acquire mass. This transition is not associated with symmetry breaking and there is no obvious local order parameter.