  Context: The Rossiter-McLaughlin (RM) effect, a rotational effect in
eclipsing systems, provides unique insight into the relative orientation of
stellar spin axes and orbital axes of eclipsing binary systems.
  Aims: Our aim is to develop a robust method to analyze the RM effect in an
eclipsing system with two nearly equally bright components. This gives access
to the orientation of the stellar rotation axes and may shed light on questions
of binary formation and evolution.
  Methods: High-resolution spectra have been obtained both out of eclipse and
during the primary and secondary eclipses in the V1143Cyg system, using the
high-resolution Hamilton Echelle Spectrograph at the Lick Observatory. The
Rossiter-McLaughlin effect is analyzed in two ways: (1) by measuring the shift
of the line center of gravity during different phases of the eclipses and (2)
by analysis of the line shape change of the rotational broadening function
during eclipses.
  Results: The projected axes of both stars are aligned with the orbital spin
within the observational uncertainties, with the angle of the primary rotation
axis beta_p=0.3+-1.5 deg, and the angle of the secondary rotation axis
beta_s=-1.2+-1.6 deg, thereby showing that the remaining difference between the
theoretical and observed apsidal motion for this system is not due to a
misalignment of the stellar rotation axes. Both methods utilized in this paper
work very well, even at times when the broadening profiles of the two stars
overlap.[abridged]
