Well, conceptually, battle in D&D is "real time" ... I think they called it "game time" up until 3.5 edition when they took it out and made the complete switch to being turn based.
But if you look in the PHB (even 5e), everything is based on units of time. Combat is broken into rounds, each round is 6 seconds of game time (5e). You can have 1, or you can have 100 players acting in that 6 second time frame. Movement is calculated in rounds. Spell durations are based on rounds. The game is actually based on a conceptual real-time framework. We take turns when we play the game as a table top board game because of physical limitations of running a game, not because the game is turn based. I would argue that the game BECAME turn based under Wizards to make it easier for non-gamers to grasp.
Think of how your game battles would translate into a story - look at how novels like Dragonlance read, which were actual game sessions before publishing (different from the RA Salvatore Drizzt books which were just flat fiction). It actually happens in real time. I have always thought it was a silly sort of supsension of disbelief to sit and pick my nose while multiple enemies took turns doing things that I couldn't react to.
It is the REACTIVE nature of combat where real time shines. rather than being chess (which D&D is not), where you strategise multiple paths of movement and eliminate choice with each opponent turn, you have to react in real time to the unfolding combat.
I will absolutely concede that as a technology, RTwP is still young and has had many flaws in the past (such as pathing errors, or huge diablo like trash fights "just cuz"). But, in my estimation, RTwP is far closer to the 'game world' experience of what DND is.
I do concede as well that, when we talk about multiplayer, TB is far easier to implement.