Anyone know of any completely flavorless syrups?
A syrup is by definition a thick sweet liquid made using sugar. If you are looking to make a thick savory liquid, perhaps you want to look into thickening agents. Starches and plant-based gums are the most common thickening agents. Some examples include:


Starches: arrowroot, cornstarch, potato starch, tapioca
Plant-base gums: guar gum, xanthum gum, alginin


Each of these will have its own unique qualities so you might want to research further into which meets your need best.

The two thickener that will most likely result in a clear, "smooth" thickness that is slightly similar to a syrup would be likely gelatin or pectin. However both of these will be much thicker at cooler temperatures. So depending on your application you will need to alter the concentration. Meaning, if your savory liquid is suppose to be consumed at room temperature or cooler, make sure not to added too much of gelatin or pectin. Try to test the thickness of your savory solution by taking a small sample aside and chilling it to room temperature.