As we all await the glorious 50th anniversary special in November, I had a bit of a thought of what would happen if Rose met River. So I put it into the context. I think it was "confirmed" that Alex Kingston would not be in it but hey, that's what fanfics are for. So please enjoy how I think this encounter might go.
And there they all were facing one another.
The Doctor and Clara on one side, The Doctor and Rose on another, and River Song on the third.
"Oh, um, hello..." Rose's Doctor stumbled on the words. He still didn't know who she really was.
"You know her?" Rose asked.
Clara's Doctor answered, "Yes. Well, no. Well, not yet. In a way. I know her better."
"Much much better," River added. The Doctor grinned back at her.
Rose's Doctor grimaced, turning to Rose to explain, "I met her once. She di-"
"Ah, ah! Spoilers!" The older Doctor and River interrupted.
Clara glanced at the both of them, "Good God, you too are a couple!"
"A couple?" Rose questioned.
"Uh, um..." her Doctor stuttered, unsure how to explain.
Clara's Doctor took over, "I think I'll explain. Rose, this is River Song. River, this is Rose Tyler."
River smiled curiously at Rose, "You're Rose Tyler! I should have known! The only blonde he ever talks about. And this must be your previous regeneration then?" The last question was for the Doctor.
"Spoilers," he answered with a wink. "Ow!" He exclaimed as he rubbed his arm where Clara slapped him.
"That's right, it hasn't happened for her yet has it?" Rose's Doctor thought aloud.
"Nope," River answered. She looked back at Rose, "I'm the Doctor's, well, this Doctor's," she pointed her thumb at the elder Doctor, "...friend."
"A bit more than a friend," he mumbled flirtatiously. "Ow! Stop that!" Clara grinned innocently.
"Yeah, just about everyone can see that," Rose's Doctor quipped with an awkward sigh.
Rose raised an eyebrow at the other Doctor, "You were still hung up on me last we met."
"It's been a few centuries since then," he replied a bit more coldly than he wanted, "I moved on."
Rose put a hand up in surrender.
River broke the coming awkward silence, "So what are we going to do about our little Zygon friends, eh?"
"Right!" Both Doctors said at once. They stared at each other. Clara slapped her Doctor again.
He turned to face her, "Would you stop that?"
"You'd better stop," River suggested, "He tends to get flustered." Clara raised a curious brow.
Rose coughed awkwardly. Her Doctor looked at his counterpart curiously, "You like that?"
"We're not going to talk about it!"
"I think we are talking about it."
"Oh! I quite like him. I hope I get to...travel with him," River chose and emphasised her words.
Rose scratched her head in agitated realization, "Are you...did you just flirt with yourself?"
He stammered in embarrassment, clearly mot realizing what he had just done. River giggled, then glanced curiously at her own Doctor.
Clara caught the look and slapped him once more.
A bit less River/Rose interactions than I hoped for but hey,points for Doctor flirting with himself.