Necklace splitting problem
Jelly, 18 bytes
s2ZFṢ$€E¬,L
ŒṖṖÇÞḢ

Try it online!
Not efficient - the example has 28 jewels which wont work without huge resources since this implementation's first step would be to build a list of the 227 possible partitions.
Returns a list of lists - the segments in the order to dish them out between alternate thieves. (Re the TIO output: when a list only has a single item the implicit print does not bother with the brackets, [])
How?
s2ZFṢ$€E¬,L - Link 1, get (isUnfair, Slices): A possible partition
s2          - split into slices of length 2 (any odd one on it's own at the end)
  Z         - transpose (first item is one thief's slices, second is the others)
     $€     - last two links as a monad for €ach
   F        -     flatten
    Ṣ       -     sort
       E    - equal? (theif1's jewels == theif2's jewels)
        ¬   - not
          L - length (number of slices in the partition)
         ,  - pair

ŒṖṖÇÞḢ - Main link: necklace
ŒṖ     - all partitions
  Ṗ    - pop, we must remove the rightmost one...
              because Link 1 will say it is fair, and it will have length 1!
              (a list of one thing has all entries equal)
    Þ  - sort by
   Ç   -     last link (1) as a monad
     Ḣ - head (get the first one, i.e. minimal isUnfair, then minimal length)