  We develop a systematic approach to boundary conditions that break bulk
symmetries in a general way such that left and right movers are not necessarily
connected by an automorphism. In the context of string compactifications, such
boundary conditions typically include non-BPS branes. Our formalism is based on
two dual fusion rings, one for the bulk and one for the boundary fields. Only
in the Cardy case these two structures coincide. In general they are related by
a version of alpha-induction. Symmetry breaking boundary conditions correspond
to solitonic sectors. In examples, we compute the annulus amplitudes and
boundary states.
