  We give a survey of a couple known constructions of $p$-adic $L$-functions
including Iwasawa's construction from classical Stickelberger elements. We then
construct "real" Stickelberger elements, i.e., explicit elements in the Galois
group ring with $\mathbb{Z}_p$ coefficients that annihilate the Sylow
$p$-subgroup of the ideal class group of $\mathbb{Q}(\zeta_{p^{n+1}})^+$. In
analogy with Iwasawa's work, we show that these elements are coherent in
$\mathbb{Z}_p$-towers and give rise to twisted $p$-adic $L$-functions.
