Yes, this can happen: if $M$ is a countable $\omega$-model of ZF whose well-founded part has ordertype $\omega_1^{CK}$ (that is: has the shortest well-founded part possible for $\omega$-models), then $Ord^M$ as a linear order is just the **Harrison order**: $$\omega_1^{CK}+(\omega_1^{CK}\cdot\eta),$$ where $\eta$ is the ordertype of the rationals. This linear order does in fact have a computable copy, and is one of the basic examples/counterexamples in computable structure theory: a computable linear order which is illfounded but has no *hyperarithmetic* descending sequence (it has other nice properties too).