If we consider $ZF$ instead of $ZFC$, then we can say more.
There are examples of such results which are obtained by Krivine, using his method of realizability. For example in the paper [Realizability algebras II: New models of ZF+DC](http://www.lmcs-online.org/ojs/viewarticle.php?id=731) the following is stated:

>Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of $ZF$ and relative consistency results in set theory. We show the relative consistency of $ZF + DC$ + there exists a sequence of subsets of $\mathbb{R}$ the cardinals of which are strictly decreasing + other similar properties of $\mathbb{R}$. 

As it is stated in the introduction of the paper: 
>These results seem not to have been previously obtained by forcing.