  We construct a natural prequantization space over a monotone product of a
toric manifold and an arbitrary number of complex Grassmannians of 2-planes in
even-dimensional complex spaces, and prove that the universal cover of the
identity component of the contactomorphism group of its total space carries a
nonzero homogeneous quasi-morphism. The construction uses Givental's nonlinear
Maslov index and a reduction theorem for quasi-morphisms on contactomorphism
groups previously established together with M. Strom Borman. We explore
applications to metrics on this group and to symplectic and contact rigidity.
In particular we obtain a new proof that the quaternionic projective space,
naturally embedded in the Grassmannian of 2-planes in a 2n-dimensional complex
space as a Lagrangian, cannot be displaced from the real part of the complex
Grassmannian by a Hamiltonian isotopy.
