In the Star Wars: Republic series, multiple Jedi have red lightsabers. Given the association between red lightsabers and the Sith, why is this? I'm looking for both the in-universe and out-of-universe reasons.
I haven't heard of the Star Wars: Republic series, but I do know that the color of the lightsaber doesn't necessarily denote whether you are a Dark side user or a Light side user. It depends on you. A Jedi could have a red lightsaber. That doesn't mean he is a Dark side user. He could have gotten it from a Sith he defeated in battle if the Sith was using a red saber. Just like you can take a blue lightsaber even if you are evil, like with General Grievous. Also, artificial kyber crystals usually are red and produced a red colored lightsaber blade, but a Jedi could use an artificial kyber crystal if they wanted, but still be a Light side user.
Star Wars: Republic takes place in Legends continuity, which is simply the name for the Star Wars Expanded Universe portion (comics, books, games, older cartoons, etc.) of Lucasfilm that was produced prior to Disney purchasing them in 2012 and prior to Disney rebooting the Expanded Universe in 2014. As such, the in-universe explanation for lightsabers will differ somewhat from what's been established in the current canon of the Star Wars Universe.
Both continuities have established that lightsabers blades are generated as a result of their internal Kyber Crystals, a rare mineral found on only a few planets that were kept a closely guarded secret by the Jedi. But in Legends, the concept of synthetic crystals was introduced to explain how people who didn't have access to Kyber Crystals could also build lightsabers and, ostensibly, to explain why Sith lightsabers were always red. The in-universe explanation was that the process of forging these crystals commonly caused a red hue, but that other colors were possible. This was also the explanation given for how Luke built his green lightsaber in Return of the Jedi.
Legends continuity also established that other types of crystals could be used to build lightsabers besides Kyber. However, after Disney's Lucasfilm started rebuilding the background information for the Star Wars Universe, naturally occurring Kyber Crystals became the exclusive crystal for lightsaber construction, with the canon explanation for red lightsabers being the near-sentient Kyber Crystals "bleeding" due to being forced to be in service to the Dark Side. Synthetic Kyber Crystals have also been reintroduced into canon, but are highly unstable and explosive.
I cannot find any explicit in-universe explanation for why Adi Gallia wielded a red lightsaber in Legends, but the implicit explanation is that she used a synthetic crystal because it was generally more powerful. In rare instances, a synth-crystal powered lightsaber could "break" the blade of another simply by overpowering it.
The Jedi believed that the Sith were defeated and the last of the Dark Jedi had fallen centuries ago. Some Jedi were beginning to carry red lightsabers because the Sith and Dark Jedi were defeated, and many Jedi thought it was acceptable to wield red blades, as depicted in Star Wars: Republic. Star Wars: # 14 Emissaries to Malastare 2 of 6.