Recall that a real $r$ is set-generic over $L$ if there is a constructible forcing notion $\mathbb{P}$ and some $L$-generic filter $G\subset\mathbb{P}$ such that $r \in L[G]$.

I know that Jensen's coding the universe is a (very powerful and complicated) method to produce (class-generic) reals that are *not* set-generic over $L$.
This is the only method I've heard of to produce such reals, but I also feel that invoking Jensen's coding to show the consistency, relative to $\mathsf{ZF}$, of a non-set-generic-over-$L$ real should be overkill. So my question is:

- Is there an "easy" (that is, easier than Jensen's coding or, at least, that uses only a relatively small fragment of the overall coding machinery) proof of the consistency, relative to $\mathsf{ZF}$, of $\mathsf{ZFC}+$"There is a real which is not set-generic over $L$"?

Thanks