Frank,
I think it comes down to hardness values. NACE MR-0175, paragraph 3.5, accepts 316L provided the hardness value is 22 HRC max in the annealed condition. As per Section II Part A, 316L has a maximum hardness value of 217 Brinell or 95 Rockwell B.
The problem here is these are different methods of measuring hardness and I don't know how these values convert and compare. Methinks the maximum range set out by ASME is higher (or vice versa?) than the max range set out by NACE, so the safest bet would be to order your 316L material and specify the maximum hardness to 22 HRC as per the NACE document.
If you find a resource for converting these hardness values, let me know, the last part of my post is speculation.