I've never been in this situation, but here is my opinion. I would take the fork out, or have the store do it, and inspect things. You can check if your bearings feel gritty, which indicates wear. If not the bearings, I would still want to know what the issue is. It's probably not actually the grease inside the bearings that's leaking out, but please read further.
You have sealed cartridge bearings in your headset, like almost all bikes do these days. That means that the bearing has a rubber seal that keeps most contaminants out. The bearings are shipped with grease, and the seal keeps that inside.
Also, cartridge bearings are designed to be used until worn, and then replaced (unless your manual explicitly says otherwise; for example, Chris King and Kogel ask this for their bearings in general, Campagnolo asks this for their crankset bearings). You can open the seals carefully, e.g. with a pick, and flush out the grease and add new grease. This won't help in most cases, unless the grease has been contaminated with dirt and the bearing is still usable, or if the grease got washed out. For the latter, I expect that you would need to subject the bearing to a lot of water - e.g. a lot of power washing, or you did a river crossing, or left the bike outside in the rain a lot.
The mechanics would have greased the outside of the bearings during installation. It could be that this grease got dirty. Better to clean and regrease if that's the case. If not that, then I would still want to see what could possibly be leaking.
You can check what happens when you rotate the fork. Listen closely and check how it feels. If there is a gritty sensation, the bearings are probably worn out - the bearings and/or races probably got pitted, which dirt can accelerate. It's possible for you to hear a whirring noise but not have it feel really gritty - I think that means a dry bearing. If the bearing feels notchy or indexed, e.g. it turns with resistance and then it feels like it hits a stop somewhere, that's often caused by excessive preload (i.e. you tightened the headset too much). I don't expect the latter in this case. If you are unsure what you feel, you could try to repeat this test on a friend's bike that you know hasn't been exposed to the elements. If still unsure, then definitely take that in to a store.
Last, the color of the discharge is consistent with something rusting. It should take quite a bit of moisture exposure to get the bearing races to rust, but that might be happening. I'm not sure what else in there could be rusting. The shift and brake cable housings do contain steel, and I suppose they could be compromised and rusting.