In this paper we analyse the well-posedness of the Cauchy problem for a
rather general class of hyperbolic systems with space-time dependent
coefficients and with multiple characteristics of variable multiplicity. First,
we establish a well-posedness result in anisotropic Sobolev spaces for systems
with upper triangular principal part under interesting natural conditions on
the orders of lower order terms below the diagonal. Namely, the terms below the
diagonal at a distance $k$ to it must be of order $-k$. This setting also
allows for the Jordan block structure in the system. Second, we give conditions
for the Schur type triangularisation of general systems with variable
coefficients for reducing them to the form with an upper triangular principal
part for which the first result can be applied. We give explicit details for
the appearing conditions and constructions for $2\times 2$ and $3\times 3$
systems, complemented by several examples.