We use a homogeneous catalogue of 42 000 main-sequence wide binaries identified by Gaia to measure the mass ratio distribution, p(q), of binaries with primary masses 0.1 < M₁/M_⊙ < 2.5, mass ratios 0.1 ≲ q < 1, and separations 50 < s/au < 50 000. A well-understood selection function allows us to constrain p(q) in 35 independent bins of primary mass and separation, with hundreds to thousands of binaries in each bin. Our investigation reveals a sharp excess of equal-mass `twin' binaries that is statistically significant out to separations of 1000-10 000 au, depending on primary mass. The excess is narrow: a steep increase in p(q) at 0.95 ≲ q < 1, with no significant excess at q ≲ 0.95. A range of tests confirm the signal is real, not a data artefact or selection effect. Combining the Gaia constraints with those from close binaries, we show that the twin excess decreases with increasing separation, but its width (q ≳ 0.95) is constant over 0.01 < a/au < 10 000. The wide twin population would be difficult to explain if the components of all wide binaries formed via core fragmentation, which is not expected to produce strongly correlated component masses. We conjecture that wide twins formed at closer separations (a ≲ 100 au), likely via accretion from circumbinary discs, and were subsequently widened by dynamical interactions in their birth environments. The separation-dependence of the twin excess then constrains the efficiency of dynamical widening and disruption of binaries in young clusters. We also constrain p(q) across 0.1 ≲ q < 1. Besides changes in the twin fraction, p(q) is independent of separation at fixed primary mass over 100 ≲ s/au < 50 000. It is flatter than expected for random pairings from the initial mass function but more bottom-heavy for wide binaries than for binaries with a ≲100 au.