  We generalise the notion of a Barge-Diamond complex, in the one-dimensional
case, to a mixed system of tiling substitutions. This gives a way of describing
the associated tiling space as an inverse limit of Barge-Diamond complexes. We
give an effective method for calculating the \v{C}ech cohomology of the tiling
space via an exact sequence relating the associated sequence of substitution
matrices and certain subcomplexes appearing in the approximants. As an
application, we show that there exists a system of three substitutions on two
letters which exhibit an uncountable collection of minimal tiling spaces with
distinct isomorphism classes of \v{C}ech cohomology.
