Oily film out of exhaust

Un-burned oil or fuel, weeping out.
If you stick your finger in it, is it dark or smells like fuel?
How much oil are your engines using?
What engines?
What color is your exhaust smoke when idling to the dock?

The only down hill source I can think of is a leaking turbo.
Next would be leaking fuel coolers.

I wouldn't rule out leaking injector(s) either.

I was thinking while the engines are sitting, not running, the fuel would not be able to leak up hill to the riser, thru the turbo then down out the exhaust.
But, oil could leak from the turbine shaft then drain down hill.
I'm assuming you kids think while idling up to the dock, leaking injectors, over fueling the cylinder, would blow out with the exhaust gases, mix with the exhaust water and be the last to drain out of the transom.
That could work also.
Hopefully the OP can answer my questions and help tighten our theories.

Question to all, Does it look like an oil film coming out of his exhaust? Or just coming around the boat from somewhere else?

How much oil are your engines using?
What engines & HP tune?

Although difficult to tell just from photos, but appears to be diesel vs lube oil. Lube oil tends to stay together more and form circular pattern. Diesel is lighter and thinner and usually leads to the ribbon type streaking you are seeing.

Good point, CR.
For some reason, I had in mind that it happened upon cold start, rather than after shutdown (as the OP actually said).
On the other hand, if I understand correctly his post #7, he did experience the same leak also upon cold start, which is indeed a scenario that might point to injectors.
But I agree with the OP that it would be a weird coincidence if it never happened in the past, and out of the blue it happens on both engines...

I'm going to go out on a limb here and call it on the injectors or fuel cooler. Buildup in the exhaust water during idle running during docking, etc, leaves residue in the exhaust that slowly washes out in a manner where a little goes a very long way. Let the boat sit, and it completes its discharge. Fire it up again, and you're re-introducing the unburned fuel to the exhaust water. So, bad low idle spray or drip from an injector that produces unburned fuel through the valves, or some small amount of fuel weeping within the fuel coolers. Fuel coolers seems much less likely to be happening on both engines at the same time. So injectors...

I to have experienced this with cold temperatures with my MAN 820s. I was told not to worry about it. I will say I don’t believe I ever saw the same amount of “sheening” as the OP’s pictures show.

Lots of idling to get in your slip?
Was there any smell?
Fuel injected or carbs?

Keep us up on what you find.
Lunderic had a great idea on clutch cooler.
Oil labs may help also.
In dealing with LBCs with Q-jet carbs long ago, it just may be un-burned gas out the pipes.
Could use some pictures with the correct light reflecting on it .
Maybe dip it and smell it,, Gas or other.

GM q-jets were the fastest (cheapest) carb to pour gas down into an engine. In doing so, idle was rich.
If it is a gas sheen and you want it resolved, consider the later Holley Spread bore carbs.
This may be bux$ but may help.

IMO, if not clutch oil and is gas, your running well, ignore it.