  We construct a symplectic integrator for non-separable Hamiltonian systems
combining an extended phase space approach of Pihajoki and the symmetric
projection method. The resulting method is semiexplicit in the sense that the
main time evolution step is explicit whereas the symmetric projection step is
implicit. The symmetric projection binds potentially diverging copies of
solutions, thereby remedying the main drawback of the extended phase space
approach. Moreover, our semiexplicit method is symplectic in the original phase
space. This is in contrast to existing extended phase space integrators, which
are symplectic only in the extended phase space. We demonstrate that our method
exhibits an excellent long-time preservation of invariants, and also that it
tends to be as fast as and can be faster than Tao's explicit modified extended
phase space integrator particularly for small enough time steps and with
higher-order implementations and for higher-dimensional problems.
