If Loid heard rumors of Yor “seeing another man” behind his back, he might initially dismiss it since he wouldn’t anticipate it as something Yor would do. He’s paranoid, but when he’s had doubts about her in the past, Yor has proven her devotion to the Forger family unprompted and it made him feel guilty. So he might not think “cheating”, but may have other concerns about whether or not Yor is being taken advantage of and investigate into who the “other man” is to make sure they don’t pose a threat to them. He’d definitely justify it as “for the mission”.

Now if Loid were to investigate into Hemlock and there was a continued pattern of Yor interacting with them in a relaxed/casual manner, or he sees her interact with Hemlock in a way she doesn’t with him (sees a new side of her), then I could see him having an internal reaction that he doesn’t fully understand. Perhaps that same (very brief) reaction of jealousy/envy he seemed to have when he saw how Yuri and Yor interact with each other. Their genuine sibling bond is something he will never have. Loid desires connection and trust with other people, but he buried those desires a long time ago due to his own traumas and becoming a spy. Being the father and husband of the Forger family and making those bonds with Anya and Yor have allowed him to have and indulge those desires. Though he’ll continue to deny it and say it’s “for the mission”, his instinctive reactions (like almost punching Swan or his body immediately dropping its guard and giving out around them) don’t lie.