The petcock in the tank was pretty mucky, and was leaking even when turned off. If it was pouring gas into the carbs all the time, even when parked for long periods, it must have filled up the carb bowls and spilled over into the intake manifold and eventually found its way into the engine oil, which would explain the seven litres of what looked like muddy water that came out of the oil drain plug.
The new petcock looks like a more finished thing than what was on it. Based on the questionable mechanics on the rest of the bike, I'm guessing this was just something that fit rather than the right spec part. The one on the top is the new one and the bottom one was what was on the bike. It seems odd that Honda wouldn't actually tell you what the petcock is doing by writing the position on the thing.
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With any luck I'll actually get to ride the thing up and down the driveway later this week and find out what else it might need. If it's sorted, I can focus on winter maintenance on the Tiger and do the LED turn signals I've got for both bikes next. Come spring time I'll ride it over to my local motorbike shop, Mostly Ironheads, and have them do a safety on it and then get it sorted for the road.