  This paper gives a new perspective on the theory of principal Galois orders
as developed by Futorny, Ovsienko, Hartwig and others. Every principal Galois
order can be written as $eFe$ for any idempotent $e$ in an algebra $F$, which
we call a flag Galois order; and in most important cases we can assume that
these algebras are Morita equivalent. These algebras have the property that the
completed algebra controlling the fiber over a maximal ideal has the same form
as a subalgebra in a skew group ring, which gives a new perspective to a number
of result about these algebras.
  We also discuss how this approach relates to the study of Coulomb branches in
the sense of Braverman-Finkelberg-Nakajima, which are particularly beautiful
examples of principal Galois orders. These include most of the interesting
examples of principal Galois orders, such as $U(\mathfrak{gl}_n)$. In this
case, all the objects discussed have a geometric interpretation which endows
the category of Gelfand-Tsetlin modules with a graded lift and allows us to
interpret the classes of simple Gelfand-Tsetlin modules in terms of dual
canonical bases for the Grothendieck group. In particular, we classify
Gelfand-Tsetlin modules over $U(\mathfrak{gl}_n)$ and relate their characters
to a generalization of Leclerc's shuffle expansion for dual canonical basis
vectors.
  Finally, as an application, we confirm a conjecture of Mazorchuk, showing
that the weights of the Gelfand-Tsetlin integrable system which appear in
finite-dimensional modules never appear in an infinite-dimensional simple
module.
