Recently an exact worldsheet description of strings propagating in certain
black hole microstate geometries was constructed in terms of null-gauged WZW
models. In this paper we consider a family of such coset models, in which the
currents being gauged are specified by a set of parameters that a priori take
arbitrary values. We show that consistency of the spectrum of the worldsheet
CFT implies a set of quantisation conditions and parity restrictions on the
gauging parameters. We also derive these constraints from an independent
geometrical analysis of smoothness, absence of horizons and absence of closed
timelike curves. This allows us to prove that the complete set of consistent
backgrounds in this class of models is precisely the general family of
(NS5-decoupled) non-BPS solutions known as the JMaRT solutions, together with
their various (BPS and non-BPS) limits. We clarify several aspects of these
backgrounds by expressing their six-dimensional solutions explicitly in terms
of five non-negative integers and a single length-scale. Finally we study
non-trivial two-charge limits, and exhibit a novel set of non-BPS supergravity
solutions describing bound states of NS5 branes carrying momentum charge.
