  Measurement incompatibility is one of the cornerstones of quantum theory.
This phenomenon appears in many forms, of which the concept of non-joint
measurability has received considerable attention in the recent years. In order
to characterise this non-classical phenomenon, various analytical and numerical
methods have been developed. The analytical approaches have mostly concentrated
on the qubit case, as well as to scenarios involving sets of measurements with
symmetries, such as position and momentum or sets of mutually unbiased bases.
The numerical methods can, in principle, decide any finite-dimensional and
discrete joint measurability problem, but they naturally have practical
limitations in terms of computational power. These methods exclusively start
from a given set of measurements and ask whether the set possesses
incompatibility. Here, we take a complementary approach by asking which
measurements are compatible with a given measurement. It turns out, that this
question can be answered in full generality through a minimal Naimark dilation
of the given measurement: the set of interest is exactly those measurements
that have a block-diagonal representation in such dilation. We demonstrate the
use of the technique through various qubit examples, leading to an alternative
characterisation of all compatible pairs of binary qubit measurements, which
retrieves the celebrated Busch criterion. We further apply the technique to
special examples of trinary and continuous qubit measurements.
