We use a homogeneous catalog of 42,000 main-sequence wide binaries identified by Gaia to measure the mass ratio distribution, p(q), of binaries with primary masses $0.1<M_1/M_{odot}<2.5$, mass ratios $0.1 lesssim q<1$, and separations $50<s/{rm AU}<50,000$. 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 1,000 to 10,000 AU, depending on primary mass. The excess is narrow: a steep increase in p(q) at $0.95 lesssim q<1$, with no significant excess at $qlesssim 0.95$. A range of tests confirm the signal is real, not a data artifact 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 ($qgtrsim 0.95$) is constant over $0.01<a/{rm AU}<10,000$. 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 ($a lesssim 100$ AU), likely via accretion from circumbinary disks, 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 $0.1 lesssim q<1$. Besides changes in the twin fraction, p(q) is independent of separation at fixed primary mass over $100 lesssim s/{rm AU} < 50,000$. It is flatter than expected for random pairings from the IMF but more bottom-heavy for wide binaries than for binaries with $alesssim$100 AU.
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