  We construct an explicit algebraic example of a subshift of finite type over
a group $\Gamma$ with an invariant Markov measure which has completely positive
sofic entropy (with respect to `most' sofic approximations) and yet does not
have a direct Bernoulli factor, because its model spaces shatter into
exponentially many clusters of sub-exponential size. The example and its
analysis are related to random low-density parity-check (LDPC) codes.
