  The possibility of the existence of states with a spontaneous interlayer
phase coherence in multilayer electron systems in a high perpendicular to the
layers magnetic field is investigated. It is shown that phase coherence can be
established in such systems only within individual pairs of adjacent layers,
while such coherence does not exist between layers of different pairs. The
conditions for stability of the state with interlayer phase coherence against
transition to a charge-ordered state are determined. It is shown that in the
system with the number of layers N\leq 10 these conditions are satisfied at any
value of the interlayer distance d. For N>10 there are two intervals of
stability: at sufficiently large and at sufficiently small d. For N\to \infty
the stability interval in the region of small d vanishes
