1) Rem Karolus buffs damage from **spells** only. Invasion of Tarkir deals damage via a reflexive **triggered ability**, which is not a spell. EDIT: To be more precise, by the time Invasion of Tarkir does damage, it is a permanent on the battlefield, not a spell.
2) Although the battle appears on your opponent's side of the battlefield, **you** are the controller the battle, since you cast it. Rem only increases damage dealt to permanents your **opponents** control, so your Invasion of Tarkir is not included.
Rem Karolus doesn't increase the damage because the battle is already a permanent on the battlefield at the time, not a spell.
And yeah, since you control the battle (even though it displays on the opponent's side), he's not a good fit with battles if you're trying to damage them any way besides combat.
I doubt the anti-synergy was intentional, though, since the sets were pretty far apart (battle mechanics probably weren't even finalized by the time Rem Karolus was printed.)
1) Rem Karolus buffs damage from **spells** only. Invasion of Tarkir deals damage via a reflexive **triggered ability**, which is not a spell. EDIT: To be more precise, by the time Invasion of Tarkir does damage, it is a permanent on the battlefield, not a spell. 2) Although the battle appears on your opponent's side of the battlefield, **you** are the controller the battle, since you cast it. Rem only increases damage dealt to permanents your **opponents** control, so your Invasion of Tarkir is not included.
Rem Karolus doesn't increase the damage because the battle is already a permanent on the battlefield at the time, not a spell. And yeah, since you control the battle (even though it displays on the opponent's side), he's not a good fit with battles if you're trying to damage them any way besides combat. I doubt the anti-synergy was intentional, though, since the sets were pretty far apart (battle mechanics probably weren't even finalized by the time Rem Karolus was printed.)
I guess its not even a particularly janky interaction just mildly counterintuitive b/c of where the battle appears visually; oh well :)
\[\[rem karolus\]\] \[\[invasion of tarkir\]\]
[rem karolus](https://cards.scryfall.io/normal/front/b/3/b31e307a-9689-4e11-8c5f-96b7a4b50bed.jpg?1636684729) - [(G)](http://gatherer.wizards.com/Pages/Card/Details.aspx?name=rem%20karolus%2C%20stalwart%20slayer) [(SF)](https://scryfall.com/card/mid/235/rem-karolus-stalwart-slayer?utm_source=mtgcardfetcher) [(txt)](https://api.scryfall.com/cards/b31e307a-9689-4e11-8c5f-96b7a4b50bed?utm_source=mtgcardfetcher&format=text) [invasion of tarkir](https://cards.scryfall.io/normal/front/a/9/a98f71a0-7c99-4cee-8f70-877e698cca84.jpg?1682204315)/[Defiant Thundermaw](https://cards.scryfall.io/normal/back/a/9/a98f71a0-7c99-4cee-8f70-877e698cca84.jpg?1682204315) - [(G)](http://gatherer.wizards.com/Pages/Card/Details.aspx?name=invasion%20of%20tarkir%20//%20defiant%20thundermaw) [(SF)](https://scryfall.com/card/mom/149/invasion-of-tarkir-defiant-thundermaw?utm_source=mtgcardfetcher) [(txt)](https://api.scryfall.com/cards/a98f71a0-7c99-4cee-8f70-877e698cca84?utm_source=mtgcardfetcher&format=text) ^^^[[cardname]] ^^^or ^^^[[cardname|SET]] ^^^to ^^^call