  Universal topological data of topologically ordered phases can be captured by
topological quantum field theory in continuous space time by taking the limit
of low energies and long wavelengths. While previous continuum
field-theoretical studies of topological orders in $3$D real space focus on
either self-statistics, braiding statistics, shrinking rules, fusion rules or
quantum dimensions, it is yet to systematically put all topological data
together in a unified continuum field-theoretical framework. Here, we construct
the topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$)
as well as a $K$-matrix $BB$ term, in order to simultaneously explore all such
topological data and reach anomaly-free topological orders. Following the
spirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders,
we present general formulas and systematically show how the $K$-matrix $BB$
term confines topological excitations, and how self-statistics of particles is
transmuted between bosonic one and fermionic one. In order to reach
anomaly-free topological orders, we explore, within the present continuum
field-theoretical framework, how the principle of gauge invariance
fundamentally influences possible realizations of topological data. More
concretely, we present the topological actions of (i) particle-loop braidings
with emergent fermions, (ii) multiloop braidings with emergent fermions, and
(iii) Borromean-Rings braidings with emergent fermions, and calculate their
universal topological data. Together with the previous efforts, our work paves
the way toward a more systematic and complete continuum field-theoretical
analysis of exotic topological properties of $3$D topological orders. Several
interesting future directions are also discussed.
