  We prove that semilinear stochastic abstract wave equations, including wave
and plate equations, are well-posed in the strong sense with an
$\alpha$-H\"{o}lder continuous drift coefficient, if $\alpha \in (2/3,1)$. The
uniqueness may fail for the corresponding deterministic PDE and well-posedness
is restored by adding an external random forcing of white noise type. This
shows a kind of regularization by noise for the semilinear wave equation. To
prove the result we introduce an approach based on backward stochastic
differential equations, differentiability along subspaces and control theoretic
results. We stress that the well-posedness holds despite the Markov semigroup
associated to the linear stochastic wave equation is not strong Feller.
