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Roy Mustang and Riza Hawkeye meet as teenagers, when Roy starts learning alchemy under Riza’s father, Berthold Hawkeye. When Roy completes his training, he asks Berthold to teach him his flame alchemy, and Berthold refuses. Shortly after, he dies. Both Roy and Riza attend the funeral, and comfort each other over their loss.
But the first real and true display of trust we see in their relationship is when Riza asks Roy if he really wants to learn flame alchemy. The secret to flame alchemy is burned onto Riza’s back, so the trust required to not only allow him to learn it, but offer to, knowing that it could scare him away, is immense.
The next time we see them together is in the Ishvalan War of Extermination.
Roy has been using this very same flame alchemy to fight as a human weapon, and Riza is working as a sniper. After a battle, Riza goes up to Roy and asks “Do you remember me?” It had been years, but she never forgot her friend. And he hadn’t forgotten her either.
After the war, they both continued to serve in the military. But even though they were subordinates, they never acted greater or lesser than the other, other than using the appropriate titles. Then, Roy asked Riza to work as his private assistant, and her response? “I will follow you into Hell if you ask me to.” Riza is completely and entirely devoted to Roy.
In the very first episode, when Roy’s gloves get wet from rain, Riza is prepared with several backup sets. She knew, or at least had a good guess, that he would either lose, destroy, or dampen one of the sets, and made sure that he wouldn’t be left helpless.
Roy and Riza are constantly caring for and protecting each other. An example we see somewhat early on is Maes Hughes’ funeral, and Roy’s iconic line, “It’s a terrible day for rain.”
Due to the aforementioned helplessness when wet, Roy equates rain with being useless and unable to help his friends. So what he means by this is, “I feel useless”. Riza is initially confused by his statement, but soon catches on and comforts him.
Another one of the best examples of this is the battle with Lust. After presumably killing him, Lust told Riza that Roy was dead, and Riza’s reaction was one of the biggest emotional outbursts she has in the entire series. She collapses to the ground, sobbing and shaking. As far as she knows, her closest ally and greatest friend is gone.
When Roy shows up, he’s burned his wounds closed and is in immense pain, and Lust is about to kill Riza with her ultimate spear. What Roy does to stop her is brutal. He burns Lust to death over and over again until her power is overwhelmed and she stops regenerating, finally crumbling to ash.
Another Royai experience involving the homunculi is when Envy takes Roy’s form. When he finds Riza, he actually does fool her at the beginning.
But then Envy makes a mistake: he says, “Stay by my side, Lieutenant.” After this, Riza pulls out a gun, fully aware that this isn’t her Colonel. But how did she know? The real Roy knows Riza so well, and vice versa, that if he were the real Roy, there would be no need to tell Riza to stay next to him. That would have already been a given.
Finally, beyond what’s canon in the universe, the mangaka herself, Hiromu Arakawa, has confirmed that they love each other deeply, and are both aware of their own and the other’s feelings.
The only reason that they aren’t openly involved in a romantic relationship is because of military regulations; if they enter a relationship, at least one of them would have to leave the army. Because of Roy’s ambitions to be Fuhrer, and Riza’s role in his plan, for as long as the anti-fraternization laws exist, the two of them can never act on their mutual love.