I'm very new to the subject of forcing. I always got the impression that with forcing we begin with say a model $M$ of a theory $\sf T+I$ and produce another model $M[G]$ that is also a model of $\sf T$ but in which $\sf I$ can fail! So, there is no difference in consistency strength between the theory we began with and the one we've ended in.
However, I've recently new of a counter example where the end looks much stronger than the beginning.
Per Mathias, lets work within a universe $V$ in which all of $\sf ZF$ is true, fix a limit ordinal $\lambda > \omega$, let $M$ be the union of all transitive sets whose intersection with $\lambda$ is strictly bounded below $\lambda$. Now $M$, a model of $\sf Z$, satisfy existence of transitive closures for all sets, and existence of $V_\alpha$ for every ordinal $\alpha$, but there exists sets in $M$ for which no rank can be assigned to!
If we force over $M$ using the Shoenfield-Kunen definitions of forcing and the partial order with exactly one element, call it $1$, then we get back the original $\sf ZF$ universe V in which we are working!
So $M$ a model of $\sf Z$ that violates $\sf ZF$, carried the seeds of $\sf ZF$ within it, that applying forcing on it can yield the full $\sf ZF$ again!
What are other examples of that phenomena? That is, using forcing to build up models of STRONGER theories?
