  We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form
closed on the full space of extended monodromy data of systems of linear
ordinary differential equations with rational coefficients. This extension is
based on the results of M. Bertola generalizing a previous construction by B.
Malgrange. We show how this 1-form can be used to solve a long-standing problem
of evaluation of the connection formulae for the isomonodromic tau functions
which would include an explicit computation of the relevant constant factors.
We explain how this scheme works for Fuchsian systems and, in particular,
calculate the connection constant for generic Painlev\'e VI tau function. The
result proves the conjectural formula for this constant proposed in
\cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate
constant factors in the asymptotics of Painlev\'e II tau function.
