A minimal example is
FullSimplify[Exp[-I*x] Sqrt[Exp[I*2 x]], Assumptions -> {x > 0}]
which will just return the same input. However, what I expect to see is \pm 1 (here \pm means plus or minus)because Sqrt[Exp[I*2 x]] can be factored out to be Exp[I*(x+n* Pi)] and further equals (\pm 1) Exp[I*x].
If that is not possible, can I write some my own simplification rules to simplify this multivalued function, which only takes one of the roots, for example Exp[I*x]?
Exp[-I*x] Sqrt[Exp[I*2 x]] // PowerExpand[#, Assumptions -> x > 0] &
givesE^(I Pi Floor[1/2 - x/Pi])
andE^(I Pi Floor[1/2 - x/Pi]) // FullSimplify[#, Assumptions -> {x > 0}] &
gives(-1)^Floor[1/2 - x/Pi]
$\endgroup$