Your initial configuration, with the two moons orbiting perfectly out of phase, is not a stable one.
Over time their orbits will change very slightly as the result of various random effects, such as asteroid strikes, which will not affect both moons identically. The moons will tug on each other in surprising ways. Just read up on the three body problem and you'll see what you're up against.
This means that one moon will start orbiting ever so slightly faster than the other, at which point they will no longer orbit in perfect antiphase. As time progresses, the angle between the two moons decreases, which will only serve to amplify any instabilities in the system. Eventually they'll reach some minimum separation, which will be when the magic happens.
Edit: a CVn pointed out, with depressing plausibility, that what will probably happen is that one of the moons will simply get ejected from the Earth-Moon system and end up in a solar orbit.
Turns out you don't have to guess too hard at this sort of thing, but can instead just go look at someone else's simulator. I haven't actually researched how plausible the one I found was, but hey: it has actual greek letters and stuff in its configuration, so it seems pretty serious to me.
Behold, my two moon simulation, using "gravity simulator". One identical copy of the moon is placed next to earth, with an argument of periapsis in opposition to the regular moon. As anticipated, the moons slowly end up with their separating angle reducing, at which point one of them simply gets booted out into solar orbit. No earth shattering kaboom. Physics is so disappointing, but I console myself that it is just a simulation and in real life it might have been so much more dramatic.
You should fiddle with the simulator yourself, if you have sufficient patience; it is somewhat user-unfriendly, but workable. If you reduce the masses of the moons by at least a factor of 10 then you end up with something that merely looks a bit chaotic (and will probably fly apart given external gravitational influences, like your gas giant). Maybe if the moons were a hundred times lighter you'd succeed in getting them to work, but at the cost of your lovely big moons providing a scenic backdrop to your planet. Maybe your complex calendar system doesn't sound so bad anymore?
Here were my previous ideas, prior to running a simulation. The observation about the size differences between the moons and their parent is probably the important bit.
In our solar system, we have the moons Janus and Epimetheus which are co-orbital. They close within about 10000km of each other and then effectively swap orbits and slowly separate again, only to meet and swap again four years later. This is the best case scenario for your moons. It does mean that you do end up with moons in different orbits, and yes, that means you're going to have funny calendars. But conjunctions are cool, especially ones you can see so closely.
On the other hand, the parent planet of Janus and Epimetheus, Saturn, is about a billion times heavier than they are, which I suspect imposes a certain amount of stability on the whole system (or at least, stabilises it over a much longer timescale). In your example, the parent planet is only a hundred times heavier or so, and that does not fill me with warm fuzzy feelings.
These final ideas are clearly wrong. but I'll leave them here for future reference, if only to discourage later readers from making the same mistakes!
I'm not a skilled enough orbital mechanic to tell you exactly what happens next, but I suspect that it will be one of two things: your moons crash into each other, resulting in a Really Exciting period of meterorite bombardment on your planet... the sort best observed from a different planet. You'll get a funky ring system for a bit, which will be nice.
The final option looks a bit like this.
Except, y'know, twice as much.