  We introduce a natural way to extend celebrated spin-cluster Monte Carlo
algorithms for fast thermal lattice simulations at criticality, like Wolff, to
systems in arbitrary fields, be they linear magnetic vector fields or nonlinear
anisotropic ones. By generalizing the 'ghost spin' representation to one with a
'ghost transformation,' global invariance to spin symmetry transformations is
restored at the cost of an extra degree of freedom which lives in the space of
symmetry transformations. The ordinary cluster-building process can then be run
on the new representation. We show that this extension preserves the scaling of
accelerated dynamics in the absence of a field for Ising, Potts, and $\mathrm
O(n)$ models and demonstrate the method's use in modelling the presence of
novel nonlinear fields. We also provide a C++ library for the method's
convenient implementation for arbitrary models.
