It is a simple consequence of AD that there are no non-principal ultrafilters on $\omega$: for $U$ an ultrafilter on $\omega$, consider the game $G_U$ where players I and II play natural numbers $x_0$ < $y_0$ < $x_1$ < $y_1$ < . . .

Let $$R_I=\lbrace 0, 1, 2, . . . , x_0\rbrace\cup\lbrace y_0+1, y_0+2, . . , x_1\rbrace\cup . . .$$ and $$R_{II}=\lbrace x_0+1, x_0+2, . . . , y_0\rbrace\cup\lbrace x_1+1, x_1+2, . . , y_1\rbrace\cup . . .$$ Then either $R_I$ or $R_{II}$ is in $U$. Say that I wins if $R_I\in U$, and II wins otherwise. Then if $U$ were non-principal, neither player can have a winning strategy, by a strategy-stealing argument.

Bringing this proof down into the AC world, we have that, for instance, projective determinacy implies that no non-principal ultrafilter on $\omega$ is projective. The hypothesis here can't be removed completely: assuming $V=L$, there are projective non-principal ultrafilters.

My question is whether this reverses. Does "every projective ultrafilter is principal" imply PD over ZFC? If not, then what are the consequences of this claim - and what is the consistency strength of ZFC+"every projective ultrafilter is principal?"