An elliptic curve $E$ over $\overline{\mathbb{Q}}$ is called a *$\mathbb{Q}$-curve* if it is isogenous (over $\overline{\mathbb{Q}}$) to all its Galois conjugates -- see Are Q-curves now known to be modular? for example.

If I take a finite Galois extension $K / \mathbb{Q}$ and an elliptic curve $E / K$ whose base-extension to $\overline{\mathbb{Q}}$ is a $\mathbb{Q}$-curve, then all the Galois conjugates $E^{\sigma}$ are also defined over $K$, but the isogenies between them might not be. Supposing $E$ to be non-CM for simplicity, then what you get instead is a $K$-isogeny from each conjugate $E^{\sigma}$ to some possibly non-trivial quadratic twist of $E$. Let me say $E$ is a **strong $\mathbb{Q}$-curve** over $K$ if it's non-CM and it's actually $K$-isogenous to all its Galois conjugates. (Clearly any $\mathbb{Q}$-curve over $K$ becomes a strong $\mathbb{Q}$-curve over some finite extension $L / K$, but I want to keep $K$ fixed here.)

It's easy to produce examples of $\mathbb{Q}$-curves which aren't strong $\mathbb{Q}$-curves, by taking a strong $\mathbb{Q}$-curve and applying a quadratic twist by an element of $K^\times / K^{\times 2}$ that's not stable under $Gal(K / \mathbb{Q})$. However, I can't find any examples which aren't of this form.

Are there $\mathbb{Q}$-curves which are

nottwists of strong $\mathbb{Q}$-curves?

(I'm chiefly interested in the case when $K$ is a real quadratic field here.)