  We formulate a new class of tensor gauge field theories in any dimension that
is a hybrid class between symmetric higher-rank tensor gauge theory (i.e.,
higher-spin gauge theory) and anti-symmetric tensor topological field theory.
Our theory describes a mixed unitary phase interplaying between gapless and
gapped topological order phases (which can live with or without Euclidean,
Poincar\'e or anisotropic symmetry, at least in ultraviolet high or
intermediate energy field theory, but not yet to a lattice cutoff scale). The
"gauge structure" can be compact, continuous, abelian or non-abelian. Our
theory sits outside the paradigm of Maxwell electromagnetic theory in 1865 and
Yang-Mills isospin/color theory in 1954. We discuss its local gauge
transformation in terms of the ungauged vector-like or tensor-like
higher-moment global symmetry. The non-abelian gauge structure is caused by
gauging the non-commutative symmetries: a higher-moment symmetry and a charge
conjugation (particle-hole) symmetry. Vector global symmetries along time
direction may exhibit time crystals. We explore the relation of these
long-range entangled matters to a non-abelian generalization of Fracton order
in condensed matter, a field theory formulation of foliation, the spacetime
embedding and Embeddon that we newly introduce, and possible fundamental
physics applications to dark matter or dark energy.
