  We develop a very general version of the hyperbola method which extends the
known method by Blomer and Br\"udern for products of projective spaces to
complete smooth split toric varieties. We use it to count Campana points of
bounded log-anticanonical height on complete smooth split toric
$\mathbb{Q}$-varieties with torus invariant boundary. We apply the strong
duality principle in linear programming to show the compatibility of our
results with the conjectured asymptotic.
