= Solution
Use a longitudinally segmented <electromagnetic calorimeter> with fine transverse position readout. The <electromagnetic shower> centroids in two separated depth layers define a line, providing <photon> pointing without assuming the longitudinal collision position. Combine this information with the reconstructed primary collision vertex from charged-particle <track reconstruction>. The two lines from that vertex to the shower positions then determine the opening angle.
For an order-of-magnitude requirement, a single-direction precision of $10\ \mathrm{mrad}$ corresponds to a transverse position uncertainty of $15\ \mathrm{mm}$ over a $1.5\ \mathrm m$ vertex-to-shower lever arm, provided the vertex is known sufficiently well. Centroid reconstruction across finely segmented cells can reach this scale. At high <collision pileup>, identifying the correct primary vertex is essential: using the wrong vertex can dominate the angular error even if the shower position is accurately measured. Converted <photons> offer an additional pointing constraint from the reconstructed $e^+e^-$ conversion tracks, but conversion is not required by the method described here.
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