  We prove strong well-posedness for a class of stochastic evolution equations
in Hilbert spaces H when the drift term is Holder continuous. This class
includes examples of semilinear stochastic damped wave equations which describe
elastic systems with structural damping (for such equations even existence of
solutions in the linear case is a delicate issue) and semilinear stochastic 3D
heat equations. In the deterministic case, there are examples of non-uniqueness
in our framework. Strong (or pathwise) uniqueness is restored by means of a
suitable additive Wiener noise. The proof of uniqueness relies on the study of
related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is
a different approach with respect to the well-known method based on the Ito
formula and the associated Kolmogorov equation (the so-called Zvonkin
transformation or Ito-Tanaka trick). We deal with approximating FBSDEs in which
the linear part generates a group of bounded linear operators in H; such
approximations depend on the type of SPDEs we are considering. We also prove
Lipschitz dependence of solutions from their initial conditions.
