  Primary and strongly primary monoids and domains play a central role in the
ideal and factorization theory of commutative monoids and domains. It is
well-known that primary monoids satisfying the ascending chain condition on
divisorial ideals (e.g., numerical monoids) are strongly primary; and the
multiplicative monoid of non-zero elements of a one-dimensional local domain is
primary and it is strongly primary if the domain is Noetherian. In the present
paper, we focus on the study of additive submonoids of the non-negative
rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are
primary monoids, and we provide conditions ensuring that they are strongly
primary. Then we study local and global tameness of strongly primary Puiseux
monoids; most notably, we establish an algebraic characterization of when a
Puiseux monoid is globally tame. Moreover, we obtain a result on the structure
of sets of lengths of all locally tame strongly primary monoids.
