  We consider $\mathrm{SU}(N)$-symmetric Ginzburg-Landau models coupled to
non-compact 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 $\mathrm{O}(M)$ model. We term
this state ${\mathbb{C}{P}}^{N-1}$-neutral phase. We also show that while
$\mathrm{SU}(N)$-symmetric Ginzburg-Landau models are not superconductors or
superfluids in the usual sense, their state in external field at sufficiently
low temperature is a vortex lattice.
