  Topological phases of matter span a wide area of research shaping fundamental
pursuits and offering promise for future applications. While a significant
fraction of topological materials has been characterized using symmetry
requirements of wave functions, the past two years have witnessed the rise of
novel multi-gap dependent topological states, the properties of which go beyond
these approaches and are yet to be fully explored. Thriving upon these
insights, we report on uncharted anomalous phases and properties that can only
arise in out-of-equilibrium Floquet settings. In particular, we identify
Floquet-induced non-Abelian braiding mechanisms, which in turn lead to a phase
characterized by an anomalous Euler class, the prime example of a multi-gap
topological invariant. Most strikingly, we also retrieve the first example of
an `anomalous Dirac string phase'. This gapped out-of-equilibrium phase
features an unconventional Dirac string configuration that physically manifests
itself via anomalous edge states on the boundary. Our results therefore not
only provide a stepping stone for the exploration of intrinsically dynamical
and experimentally viable multi-gap topological phases, but also demonstrate a
powerful way to observe these non-Abelian processes notably in quantum
simulators.
