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Does this item have any affect on a Myrmidarch's Armor Training Class Feature?
A previous FAQ has clarified that Armor Training from Magus levels and from Fighter levels stacks. However, this item only raises your fighter level for the purposes of Armor Training. The level increase is equivalent to an extra +1 on Armor Training for a Fighter. Does the Magus have an implied Fighter level of 3? Would the item increase your Magus level by 4 such that you would get Armor Training 2 at 10 instead of 14 per a regular Myrmidarch Magus?

Xaratherus |

At 8th level, a myrmidarch gains armor training, as the fighter ability. At 14th level, he gains armor training 2.
This ability replaces improved spell combat and greater spell combat.
This bright red strip of cloth, stitched with images of a cheering crowd throwing garlands toward a chariot, fits across the wearer's shoulders and then diagonally down his chest to reach his opposite hip.
The wearer treats his fighter level as 4 higher than normal for the purpose of the armor training and bravery class features.
RAW, I don't believe the item would do anything for a myrmidarch. You don't gain levels in Fighter at 8th level; you gain the armor training class ability as a Magus.
From a table perspective, if a myrmidarch Magus wanted to buy something like this, I'd let him take it, but it would be a variant that increases his effective Magus level; if he cross-classed into Fighter, it wouldn't affect his Fighter levels at all.

Xaratherus |

I can't argue with that ruling. But it seems to me that the Devs intend for there to be a positive interaction between Myrmidarch and Fighter levels/class features. The wording of the item doesn't communicate that, but the stacking example in the FAQ does.
There is interaction between them - the stacking indicates that. The text of the item, though, isn't based off your 'level' in the particular class feature; it's based off the level in the class that grants the feature (in this case, Fighter).
There's a few places where this happens, unfortunately. For example, Oracles get the spell Spiritual Weapon; it was originally a Cleric spell, and the Oracle text just refers back to the Cleric. But the Cleric ability is based on WIS, while the Oracle is a CHA-based caster, and so by FAQ\RAW the Oracle still uses her WIS.

blahpers |

Looking at the description, it appears that a non-fighter essentially got armor training and bravery as a fourth level fighter. There's no requirement listed (such as having a fighter level), unlike the robes of arcane heritage which actually has some requirements. But I suspect that the intent is that without a level in fighter, you can't treat your fighter level as four higher because you don't have one--"0" is not a valid starting point.
The FAQ on myrmidon/fighter interaction probably wouldn't help you if you have no fighter levels. It doesn't simply stack class levels--it stacks features. If you have fighter levels, and the sash increases them enough to improve your armor training feature, then the sash would help you.