  We study a generic class of fermionic two-band models under synchronized
periodic driving, i.e., with the different terms in a Hamiltonian subject to
periodic drives with the same frequency and phase. With all modes initially in
a maximally mixed state, the synchronized drive is found to produce nonperiodic
patterns of dynamical quantum phase transitions, with their appearance
determined by an interplay of the band structure and the frequency of the
drive. A case study of the anisotropic XY chain in a transverse magnetic field,
transcribed to an effective two-band model, shows that the modes come with
quantized geometric phases, allowing for the construction of an effective
dynamical order parameter. Numerical studies in the limit of a strong magnetic
field reveal distinct signals of precursors of dynamical quantum phase
transitions also when the initial state of the XY chain is perturbed slightly
away from maximal mixing, suggesting that the transitions may be accessible
experimentally. A blueprint for an experiment built around laser-trapped
circular Rydberg atoms is proposed.
