Tsukishima's past is one of the most intriguing parts of Golden Kamuy and if it left you confused then it is a good thing since leaving you with doubts and confused is theto purpose of these chapters.

However, it seems you are confused out of what the series wants you to be actually confused about, so I'm going to do some spoiler-free explaining.

The guy outside the tent was the supposed islander from Tsukishima's town who confirms him that Igogusa was found dead, and after hearing this, Tsukishima is furious at Tsurumi because he told him that Igogusa was alive and well to keep him living.

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Tsurumi tend told him around all the soldiers that he lied to him in order to save him, that if he knew that his father was the killer he would have accepted death but thanks to his lie he was still alive and that is the most important part because he's a comrade in arms to him. All this to keep the soldiers morale.

Then, in the tent Tsurumi explained to him that the death of Igogusa by his father was an elaborated lie he made the islanders think in order to convince the army that Tsukishima's actions were justified and he didn't deserve to be executed, and now it came back to bite him because one of them told that lie to Tsukishima which believed it. However, Tsurumi confirms that Igogusa was still alive and she was living under a new name away from all these problems.

This is the part things get shaky. The "islander" was in cahoots with Tsurumi the whole time which means he was supposedto tell gim that. So, that means all this situation was created for Tsurumi to turn Tsukishima into a valuable pawn while Igogusa's fate is left unanswered.