This paper contains the first algorithm that can solve disjunctions of constraints of the form $y Î B   [ f=0   Ù  g₁ ³ 0Ù...Ùg_k ³ 0 ]\exists{y}\!\in\! B \; [ f=0 \;\wedge\; g_1\geq 0\wedge\dots\wedge g_k\geq 0 ] in free variables x, terminating for all cases when this results in a numerically well-posed problem. Here the only assumption on the terms f, g ₁,..., g _n is the existence of a pruning function, as given by the usual constraint propagation algorithms or by interval evaluation. The paper discusses the application of an implementation of the resulting algorithm on problems from control engineering, parameter estimation, and computational geometry.