Consider:
--------
class AllEnum(Enum):
    @classattr
    def ALL(cls):
        members = list(cls)
        all_value = None
        if members:
            all_value = members[0]
            for member in members[1:]:
                all_value |= member
        cls.ALL = all_value
        return all_value

class Color(AllEnum, Flag):
    RED = auto()
    BLUE = auto()
    GREEN = auto()

class IntColor(AllEnum, IntFlag):
    RED = auto()
    BLUE = auto()
    GREEN = auto()

--------

The Color class works fine, but the IntColor fails.  This is due to the way the base data type and __new__ method is discovered.  If we switch the order of the bases

  class IntColor(IntFlag, AllEnum)

it works, but having to put the mixin class last is both bizarre and unworkable when the mixin should be overriding behavior.

To get the above code snippet to fail properly:
- remove the @classattr line
- prepend from enum import Enum, Flag, IntFlag, auto

While only IntColor fails with an error, Color also fails as it is not a Flag type Enum:

>>> list(Color)
[<Color.RED: 1>, <Color.BLUE: 2>, <Color.GREEN: 3>]

It should have values of 1, 2, 4.

Possibilities:

- only allow one base Enum class
  (fails with All and IntFlag)

- only allow base Enum classes that are compatible
 (so not an Enum and a Flag)
 (would require multiple base classes for the same behavior: DocEnum,
  DocFlag, etc.)

- only allow one mixin
  (before/after that one mixin the inheritance is linear)
  (same issue as above)

- only allow one non-Enum mixin, critical Enum is last one in the bases,
  previous Enums in bases are mixins
  (all sunder/dunder methods come from critical Enum, all normal methods
  and attributes are in normal base order)

- be smarter about determining/saving the correct __new__ and base data type

The last one needs to happen, and some form of the next-to-last one (hopefully allowing for more than one non-Enum mixin).

New changeset 5bdab641da0afd8aa581dfbde4f82d88d337c4b5 by Ethan Furman in branch 'master':
bpo-29577: Enum: mixin classes don't mix well with already mixed Enums (GH-9328)
https://github.com/python/cpython/commit/5bdab641da0afd8aa581dfbde4f82d88d337c4b5

New changeset 0c076caaa82a9c6596e1fe1dbe6384d53f30a1a3 by Ethan Furman in branch '3.7':
[3.7] bpo-29577: Enum: mixin classes don't mix well with already mixed Enums (GH-9328) (GH-9486)
https://github.com/python/cpython/commit/0c076caaa82a9c6596e1fe1dbe6384d53f30a1a3