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This is a very interesting discussion! I tried a couple of different variations of the original code that do not seem to encounter the bug. One variation replaces the function with a subroutine, while the other one packs b and err into a derived type which allows one to declare the function pure.

Code 
program main
   use iso_fortran_env, only: dp => real64
   implicit none

   type :: t1
      real(dp):: b(2)
      character(:), allocatable :: err
   endtype t1

   call test_fun1()

contains

   pure function fun1(a) result(t)
      real(dp), intent(in) :: a
      type(t1):: t

      t%b = 1
      if (a > 2) then
         t%err = "a bigger than 2"
         return
      endif

   end function

   subroutine sub1(a,b,err)
      real(dp), intent(in) :: a
      real(dp), intent(out) :: b(2)
      character(:), allocatable,intent(out) :: err

      b = 1
      if (a > 2) then
         err = "a bigger than 2"
         return
      endif

   end subroutine

   subroutine test_fun1()
      real(dp) :: a,b(2)
      type(t1) :: t
      character(:), allocatable :: err
      a = 3
      t = fun1(a)
      if (allocated(t%err)) then
         print*,'err is allocated'
         print*,'next line tries to print err'
         print*,t%err
      else
         print*,t%b
      endif

      a = 1
      t = fun1(a)
      if (allocated(t%err)) then
         print*,'err is allocated'
         print*,'next line tries to print err'
         print*,t%err
      else
         print*,t%b
      endif



      a = 3
      call sub1(a,b, err)
      if (allocated(err)) then
         print*,'err is allocated'
         print*,'next line tries to print err'
         print*,err
      else
         print*,b
      endif

      a = 1
      call sub1(a,b, err)
      if (allocated(err)) then
         print*,'err is allocated'
         print*,'next line tries to print err'
         print*,err
      else
         print*,b
      endif

   end subroutine

end program
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