Odd plumbing mystery

I have a standard bathroom sink setup: a hot water tap, a cold water tap and a faucet where the mixed water emerges. Everything works normally except for this: when I turn on the cold water, a little water seeps out of the HOT water tap, where the handle meets the base. It continues to seep while the cold water is running. No water seeps out of the hot water tap when I turn on the hot water, and no water seeps out of the cold water tap under any circumstances.

What could cause this?

Sounds like the faucet mixing value maybe corroded.
How old is the faucet, it maybe a Calcium buildup In the mixing value. You maybe can get a rebuild kit to replace the internal parts.

The hot water valve stem packing is degraded and needs to be replaced. May be a washer or just wrap-around packing material.

I’m a bit confused. What do you mean by a hot water tap, a cold water tap and a faucet where they’re mixed. If you have separate taps, there wouldn’t be a single faucet where they both come out. If these are two separate taps (like you’d see in the UK) which then converge to a single point, how can you tell that the hot water is leaking?

It’s important to know your setup to diagnose this. If it’s a single handled faucet, it could just be a bad ball/stem (or one of the seals or the seat or some gunk in there). If there are two separate handles, that wouldn’t apply unless it leaks all the time.

If this observation is based on the temperature you’re feeling and not something you see, that is, cold water, after a minute or two, begins to warm up, that could be a bad mixing valve. However, it can actually be a bad mixing valve anywhere in the house (or a leaking pressure balancing diaphragm in a single handled shower valve.
The reason being that if you have single handled valve elsewhere in the house that is leaking from hot to cold (more common than you’d think), it’s feeding hot water into the cold supply. The hot and cold pipes in the house are effectively connected. It’s actually very easy to diagnose, but we can get into that after we have some more details.

No, it’s not the seating washer/valve stem or else the faucet would drip hot water all the time.

It’s most likely the O-ring seal around the cartridge. You may be able to fix it temporarily by snugging it up a bit.

I would pull the cartridge clean it up and replace the seals and seating washer while you’re there. You can get a cheap set at Home Depot.

While you’re at it, do the cold side as well, it’s probably worn as well and might fail soon anyway.

Let me know if you need more instructions or I can link to a pic or possibly a video.

I think it’s the stem packing -
read the OP:

This means that when the Hot valve is closed water is leaking out of the gap between the valve stem and the nut - that water is from the other valve, which is open. When the Hot valve is open, it doesn’t leak because the valve body is backed off and compresses the packing. I have seen this behavior before.

I read it as leaking from the base of the handle but I see what you’re trying to say now. Yes, depending on the model, it could be the seal or o-ring at the top of the valve stem under the retaining nut.

Same fix though, snug up the nut on top a bit and see if it stops then replace o-rings and washer.

My thoughts when I read the OP

Thanks all! My problem now is figuring out how to get into the valve. I have this Kohler K-10272 faucet. Based on the parts diagram and this discussion, it seems like I need to unscrew the body of the bonnet from the lower escutcheon, but my attempts to do this by hand have been unsuccessful. I don’t want to mar the finish with a pliers or Channellock. Any advice on how to apply some force to the bonnet without scratching it?

Yes, wrap the tool with a cloth.

Sorry if this non-answer isn’t helpful, but since you link to a Kohler page I wonder if you contacted Kohler on-line with this question.

I was pleasantly surprised to discover that for some reason, major plumbing supplies corporations like Kohler and Moen are commendably responsive and helpful in solving problems with their products. Once I jumped through the straightforward contact-form hoops, including submitting photos of the problem, both companies not only replied but sent me replacement and repair parts free of charge.

Thanks for that tip! I contacted Kohler yesterday and they replied today, saying that they are sending me two new cartridges free of charge.