  Counterflow superfluidity in a system with $N\geq 3$ components is
distinctively different from the $N=2$ case. The key feature is the difference
between the number ($N$) of elementary vortex excitations and the number
($N-1$) of independent branches of phonon modes, that is, the number of
superfluid modes is larger than the number of ordered phase variables. We
formulate a hydrodynamic theory of this state. We show how all the dynamical
and statistical aspects of this (``Borromean") type of ordering are naturally
described by effective $N$-component theory featuring compact-gauge invariance.
We also discuss how off-diagonal intercomponent couplings convert the Borromean
supercounterfluid into a Borromean insulator, with an emphasis on the
properties of a non-trivial state with broken time-reversal symmetry.
