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An experimental setup of three coupled $\mathcal{PT}$-symmetric wave guidesshowing the characteristics of a third-order exceptional point (EP3) has beeninvestigated in an idealized model of three delta-functions wave guides inW.~D. Heiss and G.~Wunner, J. Phys. A 49, 495303 (2016). Here we extend theseinvestigations to realistic, extended wave guide systems. We place major focuson the strong parameter sensitivity rendering the discovery of an EP3 achallenging task. We also investigate the vicinity of the EP3 for furtherbranch points of either cubic or square root type behavior.

$\mathcal{PT}$-symmetric wave guide system with evidence of a third-order exceptional point