  Experiments in the recently discovered vanadium-based kagome metals have
suggested that their charge-ordered state displays not only bond distortions,
characteristic of a ``real" charge density-wave (rCDW), but also time-reversal
symmetry-breaking, typical of loop currents described by an ``imaginary" charge
density-wave (iCDW). Here, we combine density-functional theory, group-theory,
and phenomenological modeling to investigate the complex charge-ordered states
that arise from interactions between the low-energy van Hove singularities
present in the electronic structure of $A$V$_3$Sb$_5$. We find two broad
classes of mixed iCDW-rCDW configurations: triple-$\mathbf{Q}$ iCDW,
triple-$\mathbf{Q}$ rCDW order, dubbed $3\mathbf{Q}$-$3\mathbf{Q}$, and
double-$\mathbf{Q}$ iCDW, single-$\mathbf{Q}$ rCDW order, dubbed
$2\mathbf{Q}$-$1\mathbf{Q}$. Moreover, we identify seven different types of
iCDW order, stemming from the different vanadium-orbital and kagome-sublattice
structures of the two pairs of van Hove singularities present above and below
the Fermi level. While the $2\mathbf{Q}$-$1\mathbf{Q}$ states trigger an
orthorhombic distortion that breaks the threefold rotational symmetry of the
kagome lattice, the $3\mathbf{Q}$-$3\mathbf{Q}$ states induce various types of
subsidiary uniform magnetic orders, from conventional ferromagnetism to
magnetic octupolar, magnetic toroidal, and even magnetic monopolar order. We
show that these exotic orders display unique magneto-striction,
magneto-electric, and magneto-electric-striction properties that can be probed
experimentally to identify which iCDW state is realized in these compounds. We
briefly discuss the impact of an out-of-plane modulation of the charge order
and the interplay between these complex charge-ordered states and
superconductivity.
