  How to give a natural geometric definition of a covariant Poisson bracket in
classical field theory has for a long time been an open problem - as testified
by the extensive literature on "multisymplectic Poisson brackets", together
with the fact that all these proposals suffer from serious defects. On the
other hand, the functional approach does provide a good candidate which has
come to be known as the Peierls - De Witt bracket and whose construction in a
geometrical setting is now well understood. Here, we show how the basic
"multisymplectic Poisson bracket" already proposed in the 1970s can be derived
from the Peierls - De Witt bracket, applied to a special class of functionals.
This relation allows to trace back most (if not all) of the problems
encountered in the past to ambiguities (the relation between differential forms
on multiphase space and the functionals they define is not one-to-one) and also
to the fact that this class of functionals does not form a Poisson subalgebra.
