Yamamoto noticed Karin tailing him immediately. How could he not? The girl had no real training in the use of reiatsu; at the very least, she certainly didn't know how to hide it. Perhaps she didn't even know that was something she had to worry about. She was so young, after all. She had to be a rank amateur at best.
Yamamoto would not hesitate to admit that he knew very little about the female members of the Kurosaki family, despite having 2nd Squad monitor the entire family from the moment he realized how useful Ichigo Kurosaki could be. He knew that there were two of them; Karin and Yuzu, Ichigo's sisters. From what little he did know, they both had some level of spiritual sense, and were doted on by Isshin. Karin sometimes helped in purifying the hollows that came to Karakura, and she had some interaction with Captain Sakuni. It was next to impossible to find any information on the late Kurosaki matriarch Masaki Kurosaki. All that second squad had been able to find was that she had been adored by the rest of the family, and that Isshin liked to act like an idiot in front of a poster of her. Yamamoto had a lingering suspicion that a certain pair of ex-exiles were interfering with the investigation, but it wasn't nearly a big enough matter to confront them about it.
He stopped and considered. Despite their importance in the Winter War, he and Ichigo had not truly spoken before. The one time that almost resembled a conversation was when he ordered all of the officers stationed in Karakura town to return to Soul Society and to inform Kurosaki that Inoue had been branded a traitor. It was not exactly a moment that had endeered the Captain-Commander to the future savior. And certainly, if anybody knew how to speak to Ichigo Kurosaki, it would be one of his beloved sisters...
When he spoke, he raised his voice enough so that Karin would hear him, despite not facing her and being a bit of a ways from her. "You are Karin Kurosaki, correct? I am Captain-Commander Genryusai Shigekuni Yamamoto. It is a pleasure to meet you." Normally, Yamamoto would keep to being commanding, but this girl was a civilian he had never met, despite her relations. It would be incredibly rude to speak to her as he spoke normally. Just further evidence of how long it had been since he'd been out and about, as it were.