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Jim's example is good; one can easily construct examples by using elliptic fibrations. One can also make $C_1$ and $C_2$ irreducible smooth curves. Let $$S:=E\times F\rightarrow F$$ be a trivial elliptic fibration with $E$ and $F$ elliptic curves. Construct a meromorphic function $f:F\rightarrow\mathbb{P}^1$ with a double zero and a double pole. Now, construct a branched double cover $G\rightarrow F$ whose branch locus is the double zero and double pole. The base change of $S\rightarrow F$ will be a surface $$S'\rightarrow G$$ with double fibers over the branch points of $G\rightarrow F$. The difference of the set-theoretic fibers over the zero of $f$ and over the pole of $f$ is $2$-torsion under homological equivalence and $4$-torsion under algebraic equivalence.