  The advantages of using Multi-Step corrections for simulations of lattice
gauge theories with dynamical fermions will be discussed. This technique is
suited for algorithms based on the Multi-Boson representation of the dynamical
fermions as well as for the Hybrid Monte-Carlo (HMC) algorithm and variants of
the latter, like the Polynomial-HMC. Especially the latter has the power to
deal with an odd number of fermion fields--an essential feature necessary for
realistic QCD-simulations with up-, down-, and strange-quarks. In particular,
we will discuss the application of the multi-step (actually two-step)
correction technique to a PHMC updating algorithm for twisted-mass Wilson
fermions with non-degenerate fermion masses, as it was used in recent dynamical
simulations for N_f=2+1+1 fermion flavors.
