would be the title of a man who married a ruling Queen. Although it's unclear what powers they would have in Westeros, parts of the texts suggest that it would be the Queen who ruled and the King Consort would be simply that, a consort. We can look at the first Queen of Westeros, Rhaenyra, to explore the outcome.
Long before any man had reason to doubt her innocence, the question of selecting a suitable consort for Rhaenyra had been of concern to King Viserys and his council. Great lords and dashing knights fluttered around her like moths around a flame, vying for her favor.
Princess and the Queen
One of the chief pillars of support for Rhaenyra’s claim was her consort, yet Prince Daemon represented one of her greatest weaknesses as well. The prince had made more foes than friends during the course of his adventures.
However, this seemed to depend on the person who would end up being Consort to the Queen. In Westeros, Queen Rhaenyra first looked at lords and dashing knights looking for her favour. In such a case, it is likely that Rhaenyra would rule with all the power, and the consort would behave as an ordinary Queen Consort did. However, given she she married the former King's brother, her uncle, Daemon, people feared he would take control.
"Make no mistake, should Rhaenyra ever sit the Iron Throne, it will be Daemon who rules us, a king consort as cruel and unforgiving as Maegor ever was. My own head will be the first cut off, I do not doubt, but your queen, my daughter, will soon follow.”
Princess and the Queen
However, he is still titled consort, suggesting that he is still titled as the spouse of the monarch despite having taken power.
Taking another example into account, Daenerys upon her return to Westeros and Prince Quentyn's plans of becoming her consort (before he gets fried).
A hundred years ago, Daenerys Targaryen came to Dorne to make a peace. Now another comes to make a war, and my brother will be her king and consort. King Quentyn. Why did that sound so silly?
The Winds of Winter - Arianne I
While this gives us no details about how much power Quentyn would have, it reaffirms his role as consort.
In Meereen, on the other hand, the Queen seems to be in power, and on her departure the consort takes rule.
Daenerys, my queen, I will gladly wash you from head to heel if that is what I must do to be your king and consort."
A Dance with Dragons - Daenerys VI
"Your Grace, I do not like this halt. This may be some trap. The Sons of the Harpy—"
"—have been tamed," declared Hizdahr zo Loraq. "Why should they seek to harm my queen when she has taken me for her king and consort? Now help that man, as my sweet queen has commanded." He took Dany by the hand and smiled.
The above two passages suggest that the queen is in charge, albeit we see how that changes upon Daenerys' departure below.
Hizdahr zo Loraq might be his queen's consort, but he would never be his king. "If His Grace wishes for me to remove myself from court …"
"His Radiance," the seneschal corrected. "No, no, no, you misunderstand me. His Worship is to receive a delegation from the Yunkai'i...
A Dance with Dragons - The Queensguard
All kneel for His Magnificence Hizdahr zo Loraq, Fourteenth of That Noble Name, King of Meereen, Scion of Ghis, Octarch of the Old Empire, Master of the Skahazadhan, Consort to Dragons and Blood of the Harpy,"
Note that even by his own herald, he is titled Consort to Daenerys.
And now, finally, we can look at Nymeria and her consorts, which were merely counsellors and nothing more.
Though she married twice more (first to the aged Lord Uller of Hellholt, and later to the dashing Ser Davos Dayne of Starfall, the Sword of the Morning), Nymeria herself remained the unquestioned ruler of Dorne for almost twentyseven years, her husbands serving only as counselors and consorts.
The World of Ice and Fire - Ancient History: Ten Thousand Ships
In conclusion, it is rather unclear as to what powers a consort would have, but in all cases they are second to the Queen; they can however, depending on their "worth" or "rank", rule over the queen.