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.