Refractory inclusions [calcium–aluminum-rich inclusions, (CAIs)] represent the oldest Solar System solids and provide information regarding the formation of the Sun and its protoplanetary disk. CAIs contain evidence of now extinct short-lived radioisotopes (e.g.,
26
Al,
41
Ca, and
182
Hf) synthesized in one or multiple stars and added to the protosolar molecular cloud before or during its collapse. Understanding how and when short-lived radioisotopes were added to the Solar System is necessary to assess their validity as chronometers and constrain the birthplace of the Sun. Whereas most CAIs formed with the canonical abundance of
26
Al corresponding to
26
Al/
27
Al of ∼5 × 10
−5
, rare CAIs with fractionation and unidentified nuclear isotope effects (FUN CAIs) record nucleosynthetic isotopic heterogeneity and
26
Al/
27
Al of <5 × 10
−6
, possibly reflecting their formation before canonical CAIs. Thus, FUN CAIs may provide a unique window into the earliest Solar System, including the origin of short-lived radioisotopes. However, their chronology is unknown. Using the
182
Hf–
182
W chronometer, we show that a FUN CAI recording a condensation origin from a solar gas formed coevally with canonical CAIs, but with
26
Al/
27
Al of ∼3 × 10
−6
. The decoupling between
182
Hf and
26
Al requires distinct stellar origins: steady-state galactic stellar nucleosynthesis for
182
Hf and late-stage contamination of the protosolar molecular cloud by a massive star(s) for
26
Al. Admixing of stellar-derived
26
Al to the protoplanetary disk occurred during the epoch of CAI formation and, therefore, the
26
Al–
26
Mg systematics of CAIs cannot be used to define their formation interval. In contrast, our results support
182
Hf homogeneity and chronological significance of the
182
Hf–
182
W clock.