  Bose-Einstein condensation (BEC) underpins exotic forms of order ranging from
superconductivity to superfluid 4 He. In quantum magnetic materials, ordered
phases induced by an applied magnetic field can be described as the BEC of
magnon excitations. With sufficiently strong magnetic frustration, exemplified
by the system SrCu$_2$(BO$_3$)$_2$ , no clear magnon BEC is observed and the
complex spectrum of multi-magnon bound states may allow a different type of
condensation, but the high fields required to probe this physics have remained
a barrier to detailed investigation. Here we exploit the first purpose-built
high-field neutron scattering facility to measure the spin excitations of
SrCu$_2$(BO$_3$)$_2$ up to 25.9 T and use cylinder matrix-product-states (MPS)
calculations to reproduce the experimental spectra with high accuracy. Multiple
unconventional features point to a condensation of $S = 2$ bound states into a
spin-nematic phase, including the gradients of the one-magnon branches, the
presence of many novel composite two- and three-triplon excitations and the
persistence of a one-magnon spin gap. This gap reflects a direct analogy with
superconductivity, suggesting that the spin-nematic phase in
SrCu$_2$(BO$_3$)$_2$ is best understood as a condensate of bosonic Cooper
pairs. Our results underline the wealth of unconventional states yet to be
found in frustrated quantum magnetic materials under extreme conditions.
