  Orbital physics plays a significant role for a vast number of important
phenomena in complex condensed matter systems such as high-T$_c$
superconductivity and unconventional magnetism. In contrast, phenomena in
superfluids -- especially in ultracold quantum gases -- are commonly well
described by the lowest orbital and a real order parameter. Here, we report on
the observation of a novel multi-orbital superfluid phase with a {\it complex}
order parameter in binary spin mixtures. In this unconventional superfluid, the
local phase angle of the complex order parameter is continuously twisted
between neighboring lattice sites. The nature of this twisted superfluid
quantum phase is an interaction-induced admixture of the p-orbital favored by
the graphene-like band structure of the hexagonal optical lattice used in the
experiment. We observe a second-order quantum phase transition between the
normal superfluid (NSF) and the twisted superfluid phase (TSF) which is
accompanied by a symmetry breaking in momentum space. The experimental results
are consistent with calculated phase diagrams and reveal fundamentally new
aspects of orbital superfluidity in quantum gas mixtures. Our studies might
bridge the gap between conventional superfluidity and complex phenomena of
orbital physics.
