Ryu rushed to the shrine that housed the demon statue and the Dark Dragon Blade (yes, I know that both objects were destroyed in the games they appeared in, but I needed a twist to pilot the story) bursting through the doors as Kratos was about to drive his blade into Momiji's (the shrine maiden who is also Ryu's apprentice) heart. Moving quickly, Ryu pulled out his Kusari-gama and wrapped it around his wrist, pulling back. Kratos was far stronger, but Ryu was still able to hold him back.
"Release her!" He demanded, his voice filled with rage. He yanked back more on the chain with sudden force, actually getting enough pressure to cut into Kratos' skin, causing him to shout in both anger and pain, tossing Momiji to the side and swiping at Ryu, who jumped back.
"What business do you have here?" Kratos demanded.
"The demon statue and the Dark Dragon are something no one should possess. That's why they are kept here, sealed away. You could not hope to control their power." Ryu told him.
"If they give me the power to destroy the gods who follow Zeus, I care not for anything else." Kratos said.
"Such selfish arrogance. I cannot allow this!" Ryu said, drawing the dragon sword and rushing in.
Kratos swung low at Ryu's legs, but Ryu flipped over it, then used the flying swallow to sail towards Kratos and slash him deep across the chest, landing a few feet behind him. Kratos spun around, now using his Nemean Cestus, throwing incredibly strong, deadly punches at Ryu. Ryu went on the defenses, blocking with his tonfas, then did a round kick to Kratos' face, then switched to his falcon's talons. Kratos threw attacks that would punch any normal human being in half, but with the great weight of the Cestus, he was slow and clumsy with them, so Ryu was able to TOY AROUND with the former god of war, even jump on his head and jump behind him as he was throwing a punch with the Cestus, then viciously clawing at his back. Kratos screamed with rage and spun around throwing something like a haymaker causing Ryu to jump back and avoid it, he then launched out the two lion heads and swung them at Ryu, who jumped over them and used the flying swallow once again to stab Kratos in the chest a little below the neck. Due to nothing important being stabbed, Kratos was still able to continue, punching Ryu in the head, sending him to the other end of the shrine and cracking his skull. Using one of his ninpo, Ryu was able to heal the damage done to him, then pulled out the soul-stealing Kitetsu as Kratos drew the soul-stealing Claws of Hades, summoning a large group of monsters, most which Ryu could dodge, but then the sirens Kratos summoned unleashed an ear piercing scream, deafening Ryu, causing him to hold his ears, leaving him open to the charging Cerberus Kratos just summoned. Kratos took the opportunity as Ryu was getting up to try and latch onto Ryu with the claws. Reacting quickly, Ryu blocked with the sheath of Kitetsu, throwing the sword which cut a deep slash across the stab wounds on Kratos chest left by the talons. Ryu grabbed the weapon and used a different version of the flying swallow, stabbing Kratos in the heart instead of doing a flying slash (he grabs them, flips behind them and stabs them through the chest for some reason instead of the flying slash, it also depends on the weapon, then again) then he began stealing Kratos' soul. Kratos, basically being finished by the time of the stab, was in no position to fight as Ryu drained his soul and recovered from the damage Kratos' summoned demons had done him. The then let the corpse dropped and threw a ninpo fireball (I forgot the name of the technique), burning it.
"Evil is vanquished once again." Ryu said, then went to Momiji's aid.
Reasons for the outcome:
1: I had wanted to make the fight longer, but oh well, anyway, back to the fight, Ryu has far superior speed to Kratos, even if Kratos was to use the Boots of Hermes, those still move at speeds of perception (you can see the user as they run) but Ryu can move at hyper-sonic speeds, even faster in some cases, such as the flying swallow, so Kratos had no chance of keeping up. Not to mention he's got amazing reflexes and agility that puts any acrobat to shame, so evasion is as easy as breathing for him.
2: While the Nemean Cestus are immensely heavy, if you watch Kratos in action with them, it takes him great effort to swing them rather slowly and, also if you look at it, he staggers around with them clumsily as if he were drunk or like he couldn't control them, whereas with the Falcon's talons, Ryu is fast an nimble, in complete control and can strike as simply as if he were fighting bare-handed.
3: Ryu has a natural healing factor, but also, he has ninpo that can heal him on the spot, whereas Kratos does not, giving Ryu an advantage.
4: The sheath of Kitetsu was used for many things, attacking, blocking and countering attacks, so it makes sense Ryu could use it to stop the claws while he could throw the sword like a boomerang to cut Kratos.
5: Getting away from the Kitetsu's soul stealing attack is more than just resisting the soul stealing. You also have to be able to survive impalement plus shock (I'm assuming you'd go into shock if you get stabbed, why else wouldn't you be able to move?) and get away from the person holding you. Ryu has survived impalement and even in one case being sliced in half from shoulder to hip, a wound that would easily kill Kratos. And since Kratos never survived impalement, even before Ryu started sapping his soul with the Kitetsu, it was already over.
6: Now that I look at it, Ryu's strength is actually pretty close to Dante's: He blocks attacks from the goddess, who was the size of a skyscraper, destroyed another skyscraper, plus another building. And she had a giant version of the dragon sword, which HAD to weigh at least a few tons, I'd have to say 10 at minimum, plus however much force it takes to effortlessly destroy a skyscraper and a building half the size of a skyscraper. So, since Dante (according to Insieme Per La Vittoria, you can go to him for numbers) isn't much weaker than Kratos, we could say Ryu is also close to Kratos in terms of strength.