Suppose that T* is an Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA(T*) for proper forcings which preserve these properties of T*. PFA(T*) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than the first uncountable cardinal, the P-ideal dichotomy, and a restriction on the possible order types of gaps. On the other hand, PFA(T*) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.