  We investigate the issue of assisted coherence distillation in the asymptotic
limit (considering infinite copies of the resource states), by coordinately
performing the identical local operations on the auxiliary systems of each
copy. When we further restrict the coordinate operations to projective
measurements, the distillation process is branched into many sub-processes.
Finally, a simple formula is given that the assisted distillable coherence
should be the maximal average coherence of the residual states. The formula
makes the experimental research of assisted coherence distillation possible and
convenient, especially for the case that the system and its auxiliary are in
mixed states. By using the formula,\ we for the first time study the assisted
coherence distillation in multipartite systems. Monogamy-like inequalities are
given to constrain the distribution of the assisted distillable coherence in
the subsystems. Taking three-qubit system for example, we experimentally
prepare two types of tripartite correlated states, i.e., the $W$-type and
GHZ-type states in a linear optical setup, and experimentally explore the
assisted coherence distillation. Theoretical and experimental results agree
well to verify the distribution inequalities given by us. Three measures of
multipartite quantum correlation are also considered. The close relationship
between the assisted coherence distillation and the genuine multipartite
correlation is revealed.
