  The semi-device-independent framework allows one to draw conclusions about
properties of an unknown quantum system under weak assumptions. Here we present
a semi-device-independent scheme for the characterisation of multipartite
entanglement based around a game played by several isolated parties whose
devices are uncharacterised beyond an assumption about the dimension of their
Hilbert spaces. Our scheme can certify that an $n$-partite high-dimensional
quantum state features genuine multipartite entanglement. Moreover, the scheme
can certify that a joint measurement on $n$ subsystems is entangled, and
provides a lower bound on the number of entangled measurement operators. These
tests are strongly robust to noise, and even optimal for certain classes of
states and measurements, as we demonstrate with illustrative examples. Notably,
our scheme allows for the certification of many entangled states admitting a
local model, which therefore cannot violate any Bell inequality.
