Would two Earth-like planets on opposite sides of a Sol-like star at about 1 AU be a stable configuration? Would two Earth-like planets on opposite sides of a Sol-like star at about 1 AU be a relatively stable configuration? I understand that there are tadpole-like orbits and the two planets may not stay exactly opposite each other, but oscillate around those position.

But given perturbations from other planets, say a Jupiter-like one at say 5 AU, might they eventually crash into each other, or would one end up moving to a higher orbit and the other to a lower orbit?
It would not be a stable configuration, but it wouldn't be too far from the stability.

The are 5 points where a third body could have a semi-stable orbit in a 2-body system:



The Earth is the blue point. The "anti-Earth" on the other side would be in the third Lagrange point ($\rm{L}_3$).

However, this is not a stable orbit. Any small deviation, including the effect of possible fourth bodies, would change the system so, that it results yet more deviation. Only $\rm{L}_4$ and $\rm{L}_5$ are stable, and even this has some requirements (the third body can be at most $\approx$ 3.5% of the second).

Corresponding this, the space on the other side of our Sun was examined by some space probes already, and there is nothing.