Merriam-Webster states that "the deciding factor for which of these words should be used is the sound that begins the word which follows the indefinite article, rather than the letter which does." So "an RTS" or "a real-time strategy game" would both be appropriate.
As others have already said, you go with the initial sound rather than the written letters. The written word is mostly built around conveying speech, so the rules fit what you would use if reading out loud. There are plenty of cases where writing conveys other cues than verbal, but the core of it still applies to sound.
That actually applies to most punctuation as well, depending on how one defines "most".
Any initialism is going to be counted as the first letter being a word for the purpose of a/an usage, when said letters are pronounced as letters. In the case of RTS (an initialism), you wouldn't ever say it as a word, unlike RAM, which is almost always pronounced as a word and is thus is an acronym. That's the difference between those things, btw. We tend to call all of them acronyms, and that's okay, but there is a difference.
As a gamer, I’m going to assume RTS stands for Real-Time Strategy.
I read somewhere a long time ago that just because a term is in the form of an acronym, that doesn’t change what comes before it. So in this case, it’s proper to use ‘a’ rather than ‘an.’
However, I acknowledge that saying ‘a RTS’ sounds icky, so I tend to use the “incorrect,” ‘an RTS.’
I'd rather see the vowel/consonant arbitrary rule rather than the subjective 'what does pronunciation sound like.' Ultimately, written language rules are determined by the consensus of use, not the other way around. In 1k years, the pronunciation will be obscured while the rule will be deterministic and obvious.
It's nothing to do with it being an acronym, it's just because the first sound pronounced is a vowel sound. Similarly, it's "an FPS", "an ATM", and "an SUV", but "a PPV event" and "a USB stick"