  We describe a practical algorithm for computing Brauer-Manin obstructions to
the existence of rational points on hyperelliptic curves defined over number
fields. This offers advantages over descent based methods in that its
correctness does not rely on rigorous class and unit group computations of
large degree number fields. We report on experiments showing it to be a very
effective tool for deciding existence of rational points: Among a random
samples of curves over the rational numbers of genus at least $5$ we were able
to decide existence of rational points for over $99\%$ of curves. We also
demonstrate its effectiveness for high genus curves, giving an example of a
genus $50$ hyperelliptic curve with a Brauer-Manin obstrution to the Hasse
Principle. The main theoretical development allowing for this algorithm is an
extension of the descent theory for abelian torsors to a framework of torsors
with restricted ramification.
