Their initial interactions in WoU are quite different than their usual behavior towards each other.
When they first met, Alice is just full of jealousy towards Asuna all around. First off, she is absolutely pissed about the fact that someone had the audacity to check on Kirito without her permission, because she saw Kirito (as his carer) to be her own responsibility (in a sense of property too).
The second and third shock to her showed up shortly after during the camp meeting where she realized Asuna’s beauty and command/presence. Alice is quite insecure about herself despite putting up a confident facade, one of the reasons why she never had a proper relationship with Fanatio too before Fanatio gained her own confidence, and why Fanatio was able to troll her so effectively after gaining her confidence. Alice was incredibly jealous of Asuna’s beauty, so much so that her thought process was “If Asuna was to claim she was a goddess, nobody on the field would ever question such a claim due to her beauty”.
Similarly, with the commanding presence she had on the field, she was also taken aback by the fact that she could just float down from the sky and take command of the army without any issues, which only stung more now that she was the second in command after Bercouli in the Decoy Army. You can even see the aftereffects of this when she says Asuna to rest a little and let them show what they can do as Integrity Knights the next day (which she promptly follows it up by rushing into an empty field and getting captured by Vecta instead because she never surveyed the sky to realize Vecta was hanging around there like a vulture).
As for Asuna, she is rightfully annoyed by Alice’s constant onslaught of jealousy and that’s basically her only problem. Initially, Alice is just purposefully trying to be a nuisance solely for the sake of it (from what Asuna can see of course, whereas the reality is that she is just extremely jealous and wants to just enter a pissing contest with Asuna to satisfy her pride) and Asuna is just not having any of it.
Once the intensity of the War is over, everything pretty much goes back to normal. Alice is just part of the group much like everyone else. Just like everyone else, she does have her quirks too, but it’s no different than Liz or Sinon trolling someone on occasion, more of a friendly and purposeful “annoyance” than an actual nuisance.
One of the rare things the anime did good was to cut out most of Alice’s annoying behavior out of the adaptation of War of Underworld which made her a lot less unlikeable in comparison.
