  Recent excitement in the quantum spin ice community has come from the
experimental discovery of pseudospin-$1/2$ breathing pyrochlores, including
Ba$_3$Yb$_2$Zn$_5$O$_{11}$, in which inversion symmetry is broken by the `up'
and `down' tetrahedra taking different physical sizes. We show here that the
often-neglected $J_{z\pm}$ coupling between Kramers ions, in combination with
the breathing nature of the lattice, can produce an imaginary ring flip term.
This can lead to an unconventional '$U(1)_{\pi/2}$ phase', corresponding to a
maximally dense packing of visons on the lattice. Coherent dynamics persists in
all phases, together with its emergent QED description, in a manner reminiscent
of fragmentation in spinon crystals. We characterize the enlarged QSI phase
diagram and its excitations, showing that the imaginary ring flip acts both as
a chemical potential for visons and as an effective three-photon vertex akin to
strong light-matter coupling. The novel coupling causes a structured
high-energy continuum to emerge above the photon dispersion, which is naturally
interpreted as three photon up-conversion in a nonlinear optical crystal.
