Stratification of opers with apparent singularities and isomonodromic bubbling

On a compact Riemann surface of genus $g \geq 2$, we study Simpson's stratification of the de Rham moduli space of rank-2 flat connections. Upon representing flat $\mathrm{SL}_2(\mathbb{C})$-connections by branched projective structures, or equivalently by opers with apparent singularities, the stra