  If a measurement is made on one half of a bipartite system, then, conditioned
on the outcome, the other half has a new reduced state. If these reduced states
defy classical explanation -- that is, if shared randomness cannot produce
these reduced states for all possible measurements -- the bipartite state is
said to be steerable. Determining which states are steerable is a challenging
problem even for low dimensions. In the case of two-qubit systems a criterion
is known for T-states (that is, those with maximally mixed marginals) under
projective measurements. In the current work we introduce the concept of
keyring models -- a special class of local hidden state models. When the
measurements made correspond to real projectors, these allow us to study
steerability beyond T-states.
  Using keyring models, we completely solve the steering problem for real
projective measurements when the state arises from mixing a pure two-qubit
state with uniform noise. We also give a partial solution in the case when the
uniform noise is replaced by independent depolarizing channels.
