Before we get too deep here, the secondary coilpack doesn't "fire on the exhaust stroke for emissions reasons".
The secondary plugs are there to make sure all the gas gets burned, but that's both an emissions and a power thing. Putting metered gas in the engine and not burning it isn't just bad for the environment (and the cat), it's wasting gas and leaving power on the table.
Like I said, the idea that the secondary fires on the exhaust stroke comes from mixing up dual-plug and waste-spark.
Both coilpacks, primary and secondary, fire roughly together. Meanwhile, each coilpack has two coils inside, even though it serves four cylinders.
So, when one individual coil is firing, you get spark at two plugs: one in a cylinder on compression, and the other in a cylinder on exhaust. The spark in the first cylinder is the point. The spark in the second cylinder is "wasted". It isn't done for emissions purposes, it's just not a perfectly efficient ignition system.