A first-order theory<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"><mml:semantics><mml:mi>T</mml:mi><mml:annotation encoding="application/x-tex">T</mml:annotation></mml:semantics></mml:math></inline-formula>has the<italic>Schröder-Bernstein (SB) property</italic>if any pair of elementarily bi-embeddable models are isomorphic. We prove that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"><mml:semantics><mml:mi>T</mml:mi><mml:annotation encoding="application/x-tex">T</mml:annotation></mml:semantics></mml:math></inline-formula>has an expansion by constants with the SB property if and only if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"><mml:semantics><mml:mi>T</mml:mi><mml:annotation encoding="application/x-tex">T</mml:annotation></mml:semantics></mml:math></inline-formula>is superstable and non-multidimensional. We also prove that among superstable theories<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"><mml:semantics><mml:mi>T</mml:mi><mml:annotation encoding="application/x-tex">T</mml:annotation></mml:semantics></mml:math></inline-formula>, the class of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"><mml:semantics><mml:mi>a</mml:mi><mml:annotation encoding="application/x-tex">a</mml:annotation></mml:semantics></mml:math></inline-formula>-saturated models of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"><mml:semantics><mml:mi>T</mml:mi><mml:annotation encoding="application/x-tex">T</mml:annotation></mml:semantics></mml:math></inline-formula>has the SB property if and only if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"><mml:semantics><mml:mi>T</mml:mi><mml:annotation encoding="application/x-tex">T</mml:annotation></mml:semantics></mml:math></inline-formula>has no nomadic types.