Compatibility of bar end shifters and 10 spd Tiagra

Yes.

This sounds obvious but you need shifters compatible with the cable pull ratio of Shimano road derailleurs. SRAM uses a different cable pull ratio.

Any 10 speed shifter made by Shimano will of course work. I think bar end shifters appear in the 105 and above ranges.

Component manufacturers such as Tektro and Microshift also make Shimano compatible shifters. Each shifter model will specify which group set it is compatible with.

Update - The above is true for Shimano 10 speed and below except Tiagra 4700 generation. See @Nathan Knutson's answer.

With Tiagra in particular this answer is different from if your bike had any other 10-speed group.

Presently there are two different 10-speed Tiagra generations, 4600 and 4700. (So the STI from one is ST-4600 and the other is ST-4700, etc). 4600 uses the same classic actuation ratio that all pre-11-speed Shimano road groups used other than first generation Dura-Ace. 4700, however, is an oddball group. It's 10-speed but uses the new 11-speed actuation ratio, resulting in a situation where ST-4700 is currently unique in Shimano's history in terms of number of clicks (10) and amount of cable pulled per click. No other shifters interchange with ST-4700, including any of the bar-end models. So if you have 4700-level Tiagra the answer is no, there's no way of doing this. You would have to also get a new rear derailleur from any other 10-speed road group, like for example RD-5600, RD-5700, RD-6600, RD-6700, RD-7800, or RD-7900. Then it will work. (The front derailer will be friction so no issues there - otherwise it would also have a cable pull issue.)

It hasn't become clear yet whether more 10-speed groups with 4700's cable pull will be released, but I wouldn't wait for a bar-end version.

If you have a Tiagra 4700 or a GRX 10s deraileur, there are two models available from Microshift that will work.

Microshift main page

BS-A10B: right side index only

BS-T10B: right side index and friction switchable

Both have left side friction shifter.