The closure of a rational $4$-tangle is a rational knot. But is the converse true? We could tangle up even the unknot to a hopeless mess before cutting it up, and we could cut it were it "hurts most" (where we need to do a Reidemeister-$3$ to unknot).

Note that the analogous statement already for algebraic knots is "No", as it seems to me - $8_{16}$ is algebraic but has a crossing (the leftmost on KnotAtlas) that, if cut out, yields a non-algebraic $4$-tangle.

Thus: Can the closure of a non-rational tangle yield a rational knot?