I was terrible at swallowing pills for years. My inability to swallow tablets and capsules was a contributing factor to simply lying down in bed when I had a headache instead of taking some Tylenol, for example. The few times where I had an acute condition that required swallowing a pill, it would be a big chore that occasionally resulting in spitting out the pill entirely before starting over again.
When I was in college, I was diagnosed with a chronic condition with no cure. There's treatment which prevents the symptoms, but I will most likely live with the disorder for the remainder of my life. Naturally, as something relevant to this question, the treatment involves taking pills.
My parents never tried to get me over my inability to take pills in my youth. However, terminating my medication today is potentially fatal (and even if I didn't die, I'd lose my driver's license which pretty much makes life suck in the majority of the US). I learned REAL fast how to take my pills when the consequences for not taking them were worse than the PITA of taking them. After taking pills twice a day every day for years, I can practically swallow them without any drink assisting me.
I'm not saying you need your kids to be diagnosed with a life-threatening disorder in order to get them to take their pills; many adults swallow pills without ever being diagnosed with anything more serious than the flu. My point is that, if taking the pill is less cumbersome than the alternative, you've got the motivation to learn. Then, if you have practice, you become really good at it.
Does your daughter understand how bad her allergies are? You've said the pills take a long time, a lot of water, and angry tears to get the process over with... but what would happen if the pills were skipped entirely? Depending on exactly what the allergy is (you say "bad"), I can certainly visualize much worse without the antihistamine.
If you can, I would try finding a medication that is fast-acting, and permitting your daughter postpone taking the pill until she can't stand not taking it. With a fast-acting medication, this would let her get relief soon after "admitting defeat," so to speak, and giving your daughter the choice of when to take the meds would give her additional control over her life.
Of course, it's possible that she'll avoid taking the medication at all, resorting instead to things like hydrating and Kleenex. I did the same thing when I was younger, and often toughed out in order to avoid taking pills.
If she avoids taking the pills and then complains about the allergy symptoms, remind her that she can make them go away by taking her medication.
Note: This option may be difficult to implement if she needs to take the medication while at school, as many schools do not permit children to carry around drugs of any kind, forcing them to store them all at the nurse's office.
Additionally, if your daughter's allergies are bad enough that avoiding the medication is a serious threat to her health, do NOT do this. While I personally got over my own issues with taking pills while in the face of something life-threatening, I was also well into my adult years by that point, and truly able to comprehend the consequences of my actions (or inaction). If her allergies pose a health risk, make her suck it up and take those pills (suggestions from other answers may help there); I'm sure you'd rather have a crying daughter than, for example, an emergency room visit.