We study a class of conditional independence models for discrete data with the property that one or more log-linear interactions are defined within two different marginal distributions and then constrained to 0; all the conditional independence models which are known to be non smooth belong to this class. We introduce a new marginal log-linear parameterization and show that smoothness may be restored by restricting one or more independence statements to hold conditionally to a restricted subset of the configurations of the conditioning variables. Our results are based on a specific reconstruction algorithm from log-linear parameters to probabilities and fixed point theory. Several examples are examined and a general rule for determining the implied conditional independence restrictions is outlined.