Must controllers be signatories?

Questions
3

They don’t need to be observers either. Any party works. However, note that the submitting party must have seen that contract before (either by virtue of being a stakeholder or via divulgence, the latter being deprecated).

It is also worth noting that the controller … can syntax implicitly makes a party an observer, see What's the difference between controller first and choice first? - #2 by cocreature for details

Here is an example of exercising a choice C with controller b who is neither an observer nor a signatory:

module Main where

import Daml.Script

template T
  with
    a : Party
    b : Party
  where
    signatory a
    choice C : ()
      controller b
      do pure ()

template Auth
  with
    a : Party
    b : Party
  where
    signatory a, b
    choice ExC : ()
      with
        cid : ContractId T
      controller a
      do exercise cid C

test = do
  a <- allocateParty "a"
  b <- allocateParty "b"
  auth <- submitMulti [a, b] [] $ createCmd (Auth a b)
  cid <- submit a $ createCmd (T a b)
  submit a $ exerciseCmd auth (ExC cid)
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