No, you cannot get the Phantasmal Jade mount without finishing Soulwar Quest.
That's an interesting question! According to Tibia Wiki,
The taming method is to: Use 4 Spectral Horseshoes, a Spectral Horse Tack and a Spectral Saddle after completing the Soul War Quest.
This was also pointed in following best answer to a similar question: https://www.tibiaqa.com/18469/how-to-get-the-phantasmal-jade-mount
Source: https://tibia.fandom.com/wiki/Phantasmal_Jade
According with a Instagram post from TibiaWiki Br, says you must have finish the quest to be able to get the mount.
The post is wroten on Portuguese but it means: "You can not click the items if you haven't finish the quest"
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