I suppose it is always possible that the big 60th anniversary specials set for next year will include a meeting between the Tenth and Thirteenth Doctors. But in the event that such a thing doesn’t happen, well, there is this comic book story where the two met up.
It was actually rather clever.
So, the basic idea is even though it should be easy for the Doctor to meet other incarnations of him/herself, it rarely happens because of possible time paradoxes. However, when the TARDIS decides that the Doctor cannot take Graham, Yaz, and Ryan to Woodstock in 1969, there’s a reason for it. The TARDIS always drops the Doctor off where she’s needed and not necessarily where she wants to go. So yes, she is in 1969. She’s just not in Woodstock. Or New York State. Or even somewhere in America. She’s in London. And a little investigation shows she isn’t alone.
See, here’s where the clever part comes in: she arrives at the same time and place where the Tenth Doctor and Martha Jones are stuck after their first encounter with the Weeping Angels. There may be a connection, but the Doctor figures the best way to avoid a paradox is for her to try and get some answers from Martha while the fam covertly follows the Tenth Doctor around and sees if anything weird is going on.
Then again, the Thirteenth Doctor is completely incapable of being sneaky and subtle. I’m guessing the only reason Martha doesn’t figure out what’s going on sooner is because she doesn’t really know regeneration is a thing. Heck, Thirteen uses the alias “Jane Smith,” and Martha thinks that sounds like a friend of hers. Granted, this Doctor is kicking herself for not noticing Martha is smitten with her earlier incarnation, but that all plays out well.
As for the others, Yaz eventually tries to bluff her way past the Doctor before he finds the TARDIS, but she doesn’t know Time Authority regulations as well as he does.
Of course, two Doctors might mean two enemies. It’s 1969, so the Autons are up to their usual shenanigans, and the Weeping Angels are also running around and zapping people further into the past.
Now, a story like this means the two Doctors do meet, many of the standard Doctor team-ups happen as the younger one doesn’t care for the new look of the TARDIS and the two bicker a bit, though both Doctors are disappointed that they still haven’t regenerated into a ginger yet. Ten gets upset that Thirteen ditched his stuff, and she insists she just stored his suits away in another room. Odd thing to complain about since she probably couldn’t fit in his clothes anyway. However, this is a fairly short story, so it doesn’t take long for the two Doctors to work out a plan.
Step one: find the Autons’ central consciousness.
Step two: let the Weeping Angels zap it back into the past into a time before plastics, thus ending that problem.
Step three: contain the Weeping Angels in a time cage of some kind and drop them into the heart of a sun where they won’t bother anyone for a few million years.
And then everyone goes back to where they were, the Tenth Doctor well aware he won’t remember this encounter at all.
Wait, shouldn’t Martha? Then again, Thirteen did promise her she would “live” when Martha asked if she was going to die like that was the best thing in the world, and Martha honestly had a really good life after she left the TARDIS, so that may be the best answer possible.
But hey, two Doctors, and they didn’t even argue…much.