Short Answer
No, they can't. One two black holes touch, they won't split again.
I'll sketch a little formal argument which can provide some intuition, but should not be taken too seriously. Nevertheless, I believe it provides an intuitive contradiction in the situation you presented. I invite other people to comment on possible issues and I will raise a few of them myself. At the end of this answer, I'll provide a more solid answer by referring to a rigorous theorem on black hole physics.
Intuitive Approach
Let us suppose one can somehow realize the situation you mentioned, but let me add a minor thing: suppose there is also some point particle of negligible mass on the spacetime. Suppose further this particle is in the intersection of the two black holes when they are superposed. Now, once the black holes separate (assuming the situation you presented is possible), in which of them should the particle stay? It should not be able to leave either black hole, but since they are separating and the particle is in the intersection it will need to somehow choose which black hole it will stay in.
Let me provide an alternative, but very similar, argument. for a Schwarzschild black hole, once a particle is inside the black hole, it must necessarily fall into the singularity within finite proper time. More specifically, one can show (see Problem 6 of Chap. 6 of Wald's General Relativity) that the maximum lifetime of of any observer within the event horizon is $\tau = \pi M$ ($c = G = 1$), where $M$ is the black hole's mass. However, this holds for both black holes (if we assume them to be Schwarzschild), so the particle should crash into singularity $r_i = 0$ within proper time $\tau_i = \pi M_i$. Since this must happen for both black holes and they are (by assuming the situation you proposed) getting further apart, we've reached a contradiction. If the black holes do not merge, then a particle would need to be able to escape the black holes (to be fair, I think the ).
Now for some issues with these lines of thought.
- As mentioned in the comments, General Relativity is wildly non-linear. One can't simply superpose solutions. In particular, the metric of the system is far different than just "summing" two black holes and there are gonna be effects of "gravitational energy" making the gravitational field even stronger (I'm writing "gravitational energy" to refer to the non-linear effects of the Einstein Equations, even though one can't properly define gravitational energy in a sensible and local manner). This also means that we can't treat both black holes as independent and just think of a collision with each other: as they get closer, the gravitational field changes in a non-trivial way due to their interaction;
- The result I mentioned concerning the maximum lifetime within a black hole uses the Schwarzschild metric and I'm being sloppy when applying it here, since this spacetime is not even stationary. If I recall correctly, more general black holes, such as the Reissner–Nordstrom and Kerr black holes, do not necessarily have these sorts of results, as one can tell from their conformal diagrams;
- Once the black holes are together, the pair is the black hole region and, at least in principle, the rule is nothing escapes the black hole region. Since the event horizon is a global construction, we also can't really define where one of them starts and the other one ends, and hence it doesn't really make that much sense to say the particle is within both black holes at the same time (we can't even properly define which black hole is which). This also gives a different hint on why the black holes can't split. I'd say this point is a double-edged sword for the argument haha.
Rigorous Approach
I can't really provide much more intuition, so in this section I'll just provide a statement and a reference for the detailed discussions. Proposition 9.2.5 of Hawing & Ellis' The Large Scale Structure of Space-time reads (up to notation)
Let $\mathcal{B}_1(\tau_1)$ be a black hole on $\mathcal{S}(\tau_1)$. Let $\mathcal{B}_2(\tau_2)$ and $\mathcal{B}_3(\tau_2)$ be black holes on a later surface $\mathcal{S}(\tau_2)$. If $\mathcal{B}_2(\tau_2)$ and $\mathcal{B}_3(\tau_2)$ both intersect $J^+(\mathcal{B}_1(\tau_1))$, then $\mathcal{B}_2(\tau_2) = \mathcal{B}_3(\tau_2)$.
In the above, $\mathcal{S}(\tau)$ is a partial Cauchy surface for the space-time at time $\tau$, properly defined on Proposition 9.2.3. Intuitively, it is a "photograph" of spacetime at time $\tau$. A black hole at time $\tau$ is defined at p. 317 and is the usual definition one would expect: a connected component of the region from $\mathcal{S}(\tau)$ from which nothing can escape to the null infinity. $J^+$ is the causal future.
Notice that, from these definitions, once the black holes touch, they are then one black hole only, and from the proposition I mentioned it follows that it can no longer split.
Hawking & Ellis is, of course, not the only reference discussing this result. Wald also proves it as Theorem 12.2.1 on the book I mentioned above. It is also shown on p. 73 of the Lecture Notes on Black Holes by H. S. Reall.
Best Answer
At the galactic center, there is an object called Sagittarius A* which seems to be a black hole with 4 million solar masses. In 1998, a wise instructor at Rutgers made me make a presentation of this paper
http://arxiv.org/abs/astro-ph/9706112
by Narayan et al. that presented a successful 2-temperature plasma model for the region surrounding the object. The paper has over 300 citations today. The convincing agreement of the model with the X-ray observations is a strong piece of evidence that Sgr A* is a black hole with an event horizon.
In particular, even if you neglect the predictions for the X-rays, the object has an enormously low luminosity for its tremendously high accretion rate. The advecting energy is pretty "visibly" disappearing from sight. If the object had a surface, the surface would heat up and emit a thermal radiation - at a radiative efficiency of 10 percent or so which is pretty canonical.
Of course, you may be dissatisfied by their observation of the event horizon as a "deficit of something". You may prefer an "excess". However, the very point of the black hole is that it eats a lot but gives up very little, so it's sensible to expect that the observations of black holes will be via deficits. ;-)