recursive function facHelper(n, acc) result(returner)
  integer::n
  integer::acc
  integer::returner
  if (n <= 1) then
    returner = acc
  else
    returner = facHelper(n - 1, n * acc)
  endif
end function facHelper

function factorial(n)
  integer::n
  integer::factorial
  factorial = facHelper(n, 1)
end function factorial

Error: Return type mismatch of function 'fachelper' at (1) (REAL(4)/INTEGER(4))
Error: Return type mismatch of function 'factorial' at (1) (REAL(4)/INTEGER(4))

Error: Return type mismatch of function 'fachelper' at (1) (REAL(4)/INTEGER(4))
Error: Return type mismatch of function 'factorial' at (1) (REAL(4)/INTEGER(4))

program test
  implicit none
  integer :: i
  do i=1,10
    write (*,'(i2,"! = ",i8)') i, factorial(i)
  end do
contains

[cut and paste the code from your question]

end program

gfortran -std=f95 -o fac fac.f90

1! =        1
 2! =        2
 3! =        6
 4! =       24
 5! =      120
 6! =      720
 7! =     5040
 8! =    40320
 9! =   362880
10! =  3628800

program test
  implicit none
  integer :: i
  interface
     function factorial(n)
        integer, intent(in) :: n
        integer :: factorial
     end function factorial
  end interface

 do i=1,10
   write (*,'(i2,"! = ",i8)') i, factorial(i)
 end do
end program

recursive function facHelper(n, acc) result(returner)
  implicit none
  integer, intent(in) :: n, acc
  integer :: returner
  if (n <= 1) then
    returner = acc
  else
    returner = facHelper(n - 1, n * acc)
  endif
end function facHelper

function factorial(n)
  implicit none
  integer, intent(in) :: n
  integer :: factorial
  interface
     function facHelper(n,acc)
        integer, intent(in) :: n, acc
        integer :: facHelper
     end function
  end interface
  factorial = facHelper(n, 1)
end function factorial

module fact
   private :: facHelper
contains

recursive function facHelper(n, acc) result(returner)
  implicit none
  integer, intent(in) :: n, acc
  integer :: returner
  if (n <= 1) then
    returner = acc
  else
    returner = facHelper(n - 1, n * acc)
  endif
end function facHelper

function factorial(n)
  implicit none
  integer, intent(in) :: n
  integer :: factorial
  factorial = facHelper(n, 1)
end function factorial
end module fact

program test
  use fact
  implicit none
  integer :: i

  do i=1,10
    write (*,'(i2,"! = ",i8)') i, factorial(i)
  end do
end program

recursive integer function facHelper(n, acc) result(returner)

function factorial(n)
  integer::n
  integer::factorial
  integer::facHelper ! <-- Missing information now available
  factorial = facHelper(n, 1)
end function factorial

recursive function facHelper(n, acc) result(returner)
  integer::n
  integer::acc
  integer::returner
  if (n <= 1) then
    returner = acc
  else
    returner = facHelper(n - 1, n * acc)
  endif
end function facHelper

function factorial(n)
  integer::n
  integer::factorial
  integer::facHelper
  factorial = facHelper(n, 1)
end function factorial

recursive integer function facHelper(n, acc) result(returner)
  integer::n
  integer::acc
  if (n <= 1) then
    returner = acc
  else
    returner = facHelper(n - 1, n * acc)
  endif
end function facHelper

integer function factorial(n)
  integer::n
  integer::facHelper
  factorial = facHelper(n, 1)
end function factorial

gfortran --std=f95 -c ./fac.f95

recursive function tailFacHelper(n, acc) result(returner)
  integer (kind=8)::n
  integer (kind=8)::acc
  integer (kind=8)::returner
  if (n <= 1) then
    returner = acc
  else
    returner = tailFacHelper(n - 1, n * acc)
  endif
end function tailFacHelper

function tailFac(n)
  integer (kind=8)::n
  integer (kind=8)::tailFac
  integer (kind=8)::tailFacHelper
  tailFac = tailFacHelper(n, 1_8)
end function tailFac

recursive function vanillaFac(n) result(returner)
  integer (kind=8)::n
  integer (kind=8)::returner
  if (n <= 1) then
    returner = 1
  else
    returner = n * vanillaFac(n - 1)
  endif
end function vanillaFac

f2py --overwrite-signature --no-lower fac.f95 -m liboptfac -h fac.pyf;
f2py -c --f90flags=--std=f95 --opt=-O3 fac.pyf fac.f95;
f2py --overwrite-signature --no-lower fac.f95 -m libnooptfac -h fac.pyf;
f2py -c --f90flags=--std=f95 --noopt fac.pyf fac.f95;

python timer.py

*****Fortran (optimizations level 03 enabled)*****

liboptfac.vanillaFac(20)
1000000 calls
Best of ten:  0.813575983047

liboptfac.tailFac(20)
1000000 calls
Best of ten:  0.843787193298

liboptfac.tailFacHelper(20, 1)
1000000 calls
Best of ten:  0.858899831772


*****Fortran (no optimizations enabled)*****

libnooptfac.vanillaFac(20)
1000000 calls
Best of ten:  1.00723600388

libnooptfac.tailFac(20)
1000000 calls
Best of ten:  0.975327014923

libnooptfac.tailFacHelper(20, 1)
1000000 calls
Best of ten:  0.982407093048


*****Python*****

py_vanilla_fac(20)
1000000 calls
Best of ten:  6.47849297523

py_tail_fac(20)
1000000 calls
Best of ten:  6.93045401573

py_tail_fac_helper(20, 1)
1000000 calls
Best of ten:  6.81205391884