There's some kind of big royal U.N. conference and Mario and Luigi have tagged along with Peach as "security" (really because it was going to be boring). Luigi runs into Prince Peasley, who he tried desperately to get the attention of last time they met, and realized afterward that maybe Peasley wasn't interested and ignored him on purpose. (This is not true, Peasley legitimately didn't realize Luigi was into him, people act like that around him all the time!) So in a misguided attempt to not look like a loser, Luigi says he's there as a guest's romantic partner. Peasley cheerfully says he hopes he sees the two of them at some of the social events later and moves on.
Meanwhile, Bowser is actually trying to play nice and get some other countries to trade with the Koopa Kingdom, which is hard not just because of their history of invasions but also because of his personal reputation. More than a few royals are reluctant to even talk to him because they know what happened with the last princess who was polite to him. When Bowser finds out that's part of the problem, he quickly says he's over all that. In fact, he's got a consort now!
What to do, what to do. 🤔
Later, Bowser practically trips over Luigi, and decides he's as good a target as any to take out his rage on. Despite already being miserable Luigi's not having any of that, and they agree to find a private room to have a proper fight in. Neither of them wants to get too hurt, so they're only disheveled and out of breath when they emerge.
Most people don't think twice about two conference guests coming out of a room, but Peasley happens to be in the hallway, takes one look and says, "I see! Good for you!"
Luigi is too stunned to say anything, but when Bowser asks what the hell that was, Luigi explains as quickly as he can and offers to go clear things up.
"No way! This is perfect!"
Bowser drags Luigi back into the room, explains his situation, and makes a mostly-decent proposal. All they have to do is make a couple appearances at those stupid social events on the conference schedule, Peasley will think Luigi traded up (Luigi: "Sure... Up...") and the stuck-up royals will think Bowser has a nice boring boyfriend, problems solved all around!
As much as Luigi doesn't want to do it, he wants to explain the truth to Peasley even less. He agrees.
(and tells Mario and Peach immediately, of course, who think it's hilarious)
So... that's that. He's kind of dating Bowser. The first social event is a cocktail party, where everybody's supposed to play nice and make light conversion. It's incredibly awkward, but Luigi can tell Bowser is making a serious effort to keep his temper in check, no matter how many people have snide comments about him or his "date." Mostly to keep from having to fight him again, Luigi talks to him and listens to his complaints and agrees that a lot of the other royals are being jerks. It's... kind of amazing what a difference it makes in Bowser's personality just to be heard.
The next day's social event is a showcase of regional food and drinks, which was entirely voluntary but most of the royals have turned into a passive-aggressive competition. Since the Koopa Kingdom couldn't be bothered to take part, all Bowser and Luigi have to do is walk around and eat free food. They have a pretty good time, all things considered, especially since Luigi has been to a lot of these countries in his adventures and can give informed opinions.
It goes on like that for the rest of the week. Every day, Bowser and Luigi are a little more surprised by each other, a little more comfortable in each other's presence. Bowser protects Luigi several times from getting stepped on or shoved aside, and Luigi starts getting snide right back at anyone who makes a remark about Bowser's manners.
Bowser mentions some of the trouble he's having and Luigi does his best to offer advice. Without thinking about it, Bowser brings Luigi along to his next meeting and he's such a balancing force that it goes better than any meeting Bowser has had all week.
Then it's the last day, and the last event, and after this they'll be done with each other. Back to normal. Back to the way things were before. This one's a concert and a fair number of royals are dancing with their partners or each other (it's a great opportunity to make matches, after all). Mario and Peach get up to dance, and a minute later Bowser gets up and leaves the room.
They're outside in the garden, alone under the moonlight, and Luigi says he understands how Bowser feels. That he's not the only person who's ever had feelings for someone who loves someone else.
"That's not it," Bowser says.
And before Luigi can ask what it is then, Bowser picks him up by the collar and kisses him.
In the morning Luigi leaves with the Mushroom Kingdom delegation, as planned, and nobody notices that he didn't spend the night in his room. Nor does anybody ask where he's spending those days that he's not at home.
The next year, when he and Bowser announce their engagement, the guest list includes the royal families of half the kingdoms on the planet.