Many Northeastern Neo-Aramaic languages have two distinct strategies for marking perfective aspect, both morphosyntactically restricted in different ways. The canonical perfective strategy – the perfective verb base – typically bans certain types of objects, while the secondary perfective strategy – the imperfective verb base bearing the perfective prefix qam- – requires a certain type of object. This paper deals mainly with the secondary perfective strategy, attempting to understand and formally characterize its distribution and structure. While this phenomenon is well-known in the descriptive and historical literature (see Coghill 1999 and references therein), it has never before received theoretical treatment. I argue that qam- is a high perfective aspect marker analogous to certain so-called “superlexical” prefixes in Slavic languages, which have a perfectivizing function and select for an imperfective stem (Babko-Malaya 1999; 2003; Ramchand 2004; Romanova 2004; Tatevosov 2008; Gribanova 2013; i.a.). Slavic-like stacking of high aspects is thus shown to exist in languages that lack a corresponding system of rich lexical aspect (cf. the Slavic “lexical” prefixes). The secondary perfective in Northeastern Neo-Aramaic also illuminates another cross-linguistic pattern: apparent temporal or aspectual mismatches between the clause level and the verb-stem/base level, like those found in Indo-Iranian languages (Haig 2008), need not imply that the verb bases/stems lack tense and aspect. Finally, I show that the secondary perfective furnishes a new type of argument for the syntactic nature of φ-agreement.