Cardinalities of Ultraproducts of Finite Sets in ZF + DC

It is consistent with ZF + DC that there exists an ultrafilter $U$ on $\omega$ such that two infinite ultraproducts of finite sets, $\prod A_n\mathrm{/}U$ and $\prod B_n\mathrm{/}U$, have the same cardinality if and only if $0<{\lim }_U\mathrm{\mid }{A}_n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{B}_n\mathrm{\mid }<\mathrm{\infty }$. In particular, this holds in $W[U]$, where $W$ is the Solovay Model and $U$ is $[\omega {]}^{\omega }$-generic.