By Borel determinacy + Martin's cone theorem, for every countable fragment $\mathcal{A}$ of $\mathcal{L}_{\omega_1,\omega}$ there is a turing degree ${\bf c}$ such that for all ${\bf d}\ge_T{\bf c}$ we have $(\mathcal{D},{\bf c})\equiv_\mathcal{A}(\mathcal{D},{\bf d}).$ Here "$\equiv_\mathcal{A}$" denotes $\mathcal{A}$-elementary equivalence, and $(\mathcal{D},{\bf a})$ is the structure gotten from the poset of Turing degrees by adding a constant symbol naming ${\bf a}$. A key step here is the countable completeness of the Martin filter: we use the cone theorem to get a cone on which $\varphi$ is "stable" for each sentence $\varphi\in\mathcal{A}$, then take the intersection of these cones. This breaks down once we look at uncountable sets of infinitary sentences.
Question 1: Is it consistent with $\mathsf{ZF+DC}$ that there is a Turing degree ${\bf c}$ such that for all ${\bf d}\ge_T{\bf c}$ we have $$(\mathcal{D},{\bf c})\equiv_{\mathcal{L}_{\omega_1,\omega}}(\mathcal{D},{\bf d})?$$
Note that, roughly speaking, any way of associating a countable structure to a degree would have to result in isomorphic structures when applied to degrees $\ge_T{\bf c}$; for example, if ${\bf d}\ge_T{\bf c}$ then we would need to have $\mathcal{D}_{\le_T{\bf d}}\cong\mathcal{D}_{\le_T{\bf c}}$. This is a consequence of Scott's isomorphism theorem. But this doesn't actually seem that great an obstacle. (EDIT: looks like I misjudged this, see Ted Slaman's comment below.)
In fact, even a weaker question seems difficult (and to me this suggests that the answer to Question 1 should be negative):
Question 2: Is it consistent with $\mathsf{ZF+DC}$ that there are distinct Turing degrees ${\bf a},{\bf b}$ such that $$(\mathcal{D},{\bf a})\equiv_{\mathcal{L}_{\omega_1,\omega}}(\mathcal{D},{\bf b})?$$
Of course a negative answer to Question 2 can't be hoped for at the moment, since that would imply a negative answer to the automorphism question which is still inaccessible. However, a positive answer does seem not-too-impossible; moreover, obstacles to a simple positive answer to Q2 might lead to a negative answer to Q1.
To clarify, Q1 is the question I'm primarily interested in; my interest in Q2 is mostly because it might help clarify Q1. I'd also be interested in answers to the analogous questions for other degree structures (e.g. the hyperarithmetic degrees are often more tractable than the Turing degrees, and in particular they're known to be rigid so a negative answer to "Q2$^{hyp}$" might be accessible), although my primary interest is in the Turing degrees.
