We consider SU(N)-symmetric Ginzburg-Landau models coupled to a noncompact Abelian gauge field focusing on the case N > 2 at finite temperature. We show that, at least for sufficiently large gauge-field coupling constants, these models have two phase transitions. The intermediate phase between the symmetric and low-temperature phases is a state with composite neutral order and no Meissner effect. In this neutral phase the system spontaneously breaks only the symmetry associated with phase differences and density differences between components. For N > 2, in contrast to the SU(2) case, the neutral state cannot be mapped onto an O(M) model. We term this state the SU(N)(n) phase.