According to the literature, well known tariff reform rules that are guaranteed to increase welfare will not necessarily increase market access, while rules that are guaranteed to increase market access will not necessarily increase welfare. Such conflict between welfare and market access objectives of trade policy is problematic and calls for finding alternative tariff reform rules that can achieve both objectives at the same time. The present paper contributes to this aim by using a new set of tariff reforms that are based on local optimality. Using such reforms it is shown that market access and consumer welfare will always be weakly compatible, in the sense that reforms based on each objective have the same signed effect on the other objective. For strong compatibility, whereby both objectives increase as a result of a locally optimal tariff reform, we derive both a necessary and sufficient condition and a simple sufficient condition.