  Phase transitions in condensed matter are often linked to exotic emergent
properties. We study the fully frustrated bilayer Heisenberg antiferromagnet to
demonstrate that an applied magnetic field creates a novel emergent
criticality. The quantum phase diagram contains four states, the DS (singlets
on every interlayer dimer bond), DTAF (all triplets with antiferromagnetic
order), TC (a singlet-triplet checkerboard) and FM (saturated ferromagnet). The
thermal phase diagram is dominated by a wall of discontinuities extending from
the zero-field DTAF-DS transition to a quantum critical endpoint where the
field drives the DTAF and TC into the FM. This first-order wall is terminated
at finite temperatures by a line of critical points, where the
Berezinskii-Kosterlitz-Thouless (BKT) transition of the DTAF and the thermal
Ising transition of the TC also terminate. We demonstrate by quantum Monte
Carlo simulations that the BKT transition does not change the Ising nature of
the DTAF-DS critical line. By contrast, the combination of symmetries merging
on the multicritical DTAF-TC line leads to a 4-state Potts universality not
contained in the microscopic Hamiltonian, which we associate with the
Ashkin-Teller model. Our results represent a systematic step in understanding
emergent phenomena in quantum magnetic materials including the
``Shastry-Sutherland compound'' SrCu$_2$(BO$_3$)$_2$.
