  We study a lattice model comprising four massless reduced staggered fermions
in four dimensions coupled through an $SU(4)$-invariant four-fermion
interaction. We present both theoretical arguments and numerical evidence that
no bilinear fermion condensates are present for any value of the four-fermi
coupling, in contrast to earlier studies of Higgs--Yukawa models with different
exact lattice symmetries. At strong coupling we observe the formation of a
four-fermion condensate and a mass gap in spite of the absence of bilinear
condensates. Unlike those previously studied systems we do not find a
ferromagnetic phase separating this strong-coupling phase from the massless
weak-coupling phase. Instead we observe long-range correlations in a narrow
region of the coupling, still with vanishing bilinear condensates. While our
numerical results come from relatively small lattice volumes that call for
caution in drawing conclusions, if this novel phase structure is verified by
future investigations employing larger volumes it may offer the possibility for
new continuum limits for strongly interacting fermions in four dimensions.
