There are a couple of ways to approach this, and I'll try to keep my notation consistent with <PERSON>. I think the most fundamental thing to recognize is that a complex $t_1$ is a choice (so it can't really have an effect), but, in the problem described, a complex $t_2$ is inevitable.
Playing with phases
Let's say, before we apply the field, everything is time-reversal invariant and all of the hopping terms ${t_1,t_2}$ for nearest and next-nearest neighbor are (chosen to be) real. Now, this is not entirely necessary; you could for instance, change your basis by multiplying all of the B-site basis functions with a phase factor. Then the nearest-neighbor hopping $A\rightarrow B$ picks up a phase, and hopping $B\rightarrow A$ picks up the opposite phase. Voila, complex $t_1$. But this is just a game with the basis, so it clearly won't change the spectrum or have any physical consequences.
We could do something similar with next-nearest neighbor, but we would have to obnoxiously redefine our tight-binding basis function so that it varies in phase from one A-site to another A-site, and then the hopping element would not be uniform between all A-sites (or between all B-sites). It probably isn't clear a priori what restrictions math may place on us in this game, but we'll get to that. My point here is just that there's no obvious reason you can't put a (site-dependent) phase in $t_2$ before applying the field, at the cost of making the problem quite annoying. But again, it's just a basis-change, nothing physical.
Phase around a closed loop
Now let's apply a field, but keep in mind that we can still change bases (or gauges) to affect the phases of the hopping terms as discussed above. 298
As <PERSON> notes, the effect of the field is to multiply the hopping elements by a phase factor $\exp\left[ie/\hbar\int A\cdot dr\right]$. Let's try two experiments.
* First, do some nearest-neighbor (NN) hopping around some closed path and look at the phase we accumulate. If you try this out, you'll notice that, if you're NN hopping on this lattice and you want to get back to where you started, you will always enclose some integer number of hexagonal unit cells (because you can't just "cut through" a cell with only NN). And every cell has zero net flux, so your phase factor for a closed NN path is $\exp\left[ie/\hbar\oint A\cdot dr\right]=\exp\left[ie/\hbar\sum_{cells} \iint B dA\right]=\exp\left[ie/\hbar\sum_{cells} 0\right]=1$
* Now, let's do some second nearest neighbor (2NN) hopping around a closed path. For instance, let's take a loop of three sites within the same hexagon. The path enclosed is some portion of the hexagon, which may have non-zero net flux so your path can accumulate some phase $\exp\left[ie/\hbar\oint A\cdot dr\right]=\exp\left[ie/\hbar\iint BdA\right]=e^{i\theta}$
Note that changing bases or gauges does not alter either of the above values. The NN path will always collect no phase, and the same 2NN path will always collect the same $e^{i\theta}$, because we've expressed these phases in terms of base- and gauge-invariant quantities (just the magnetic field $B$).
Consequences
So, if we try to play with the complex hopping element between second nearest neighbors, we won't be able to get it back to being a real value everywhere, because we know that the 2NN hopping elements have to combine to form a phase $e^{i\theta}$ around a certain loop. On the other hand, this restriction does not prevent us from making every NN hopping element just real $t_1$ because the accumulated phase is always 0. Big picture, the topology of the paths possible with NN and 2NN are different, and restrict the allowable phases in different ways.
The complex possibilities for $t_1$ don't actually have any effect on the spectrum, and we see this because we can just transform this phase away. But the complexity of $t_2$ is something we're stuck with, so Haldane explores its consequences.
I hope that helps! 768