  We investigate N=4 SYM coupled to fundamental flavours at nonzero imaginary
quark chemical potential in the strong coupling and large N limit, using
gauge/gravity duality applied to the D3-D7 system, treating flavours in the
probe approximation. The interplay between Z(N) symmetry and the imaginary
chemical potential yields a series of first-order Roberge-Weiss transitions. An
additional thermal transition separates phases where quarks are bound/unbound
into mesons. This results in a set of Roberge-Weiss endpoints: we establish
that these are triple points, determine the Roberge-Weiss temperature, give the
curvature of the phase boundaries and confirm that the theory is analytic in
mu^2 when mu^2~0.
