The notion of ex post constrained efficiency is appropriate for efficiency considerations in environments where a desideratum is robustness to informational specifications. It is a Pareto criterion based on ex post incentive efficiency defined by Holmstrom and Myerson 1983. In the first example of an auction environment, assuming incentive compatibility, it is shown that even when there exist ex post efficient allocation rules, i.e., the second-price auction, there may exist ex post constrained efficient allocation rules that are not ex post efficient. One such allocation rule is allocating the object to a pre-specified buyer at the lowest possible price. A second example applies the notion to bilateral trade were a seller and a buyer have private value and cost over an indivisible commodity. It is shown that, contrary to main results in Hagerty and Rogerson 1987 and Copic and Ponsati 2016, allocation rules where the price paid by the buyer differs from the payment to the seller are ex post constrained efficient. Ex post constrained efficiency is the weakest welfare criterion, e.g., ex ante constrained efficient allocation rules must be ex post constrained efficient. Ex post constrained efficiency can also be defined via maximization of a social welfare criterion, or as a robust ex ante or interim welfare criterion over the individuals' rich types.