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Dry teak can be very good about soaking up oils.
I hope you get a good solution to your problem as posted. I will be hoping to be shown hot to work better.
Failing that, a light coating of a finishing oil might make the color uniform and you can put whatever finish you want over it once it dries. Essentially, you would be doing an oil stain for coloring.
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Try covering the oily spots with cornstarch. It might help draw it out. Change out the cornstarch for fresh after it has absorbed some of the oil.
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Teak is very oily to begin with, so if it's dry it's going to suck it right up.
If you're talking about bar oil, you could probably sand the entire surface and very lightly rub in a uniform coat of bar oil. That's probably the only way you're going to be able to make it uniform. Bar oil is very thick (basically 20W50 motor oil) and you aren't going to find anything that will be equivalent, at least nothing intended for wood. Given the nature of teak you might be able to have only that small amount penetrate and still be able to use a natural oil on top.