Who exploits comparative advantage in international trade? Suppose that under autarky, Colombia makes coffee and cars at \$1 and \$2 each, and Japan makes coffee and cars at \$20 and \$10 each. Since Japan has a comparative advantage in car manufacture, the standard theory says that Colombia will import cars and export coffee.

However, I'm having a hard time imagining how this actually happens in practice. If I run a Colombian car business, why would I import cars from Japan, when they cost five times as much? Why not just buy cars at home?

In general, I see no reason why any business dealing in a single market should care about price ratios; they only see the absolute prices of a single good. For the trade to actually happen in this case, you'd either need a company that trades both cars and coffee (do those actually exist?) or government-controlled trade. Neither of these feel realistic.

If we can look at Ricardo's original example, the same problem appears. Ricardo is assuming that production across an entire country is synchronized in some way; his theory would work if a single business in each country controlled all production of all goods there. But I don't see how the logic holds if there are separate companies producing cloth and wine.

In real life, how are comparative advantages actually exploited?
It's fine to make a supposition that it costs Colombia twice as much to make cars as coffee, and Japan twice as much to make coffee as cars.  It's also fine to express these ratios in terms of the respective local currencies.    

The apparent problem you describe only arises because you go beyond the above and express the costs in terms of a common currency (\$).  But in a situation of autarky, there is no way in which the exchange rates needed to convert prices in other currencies to \$ could be determined. Thus your supposition is incoherent.

The way in which comparative advantage would be exploited, given the ratios you describe and given free trade, is that the rate of exchange of Colombian to Japanese currency would adjust to a level at which, in both countries, Colombian coffee is cheaper than Japanese, and Japanese cars are cheaper than Colombian. The Colombian car business will then be able to benefit from comparative advantage while not needing to look beyond the relative prices of locally produced and imported cars. 

Addendum:  If, in the past, two countries both used gold as a currency, this would not, given autarky, be equivalent to a common currency. Even if the gold were physically the same, it might just happen (eg for geological reasons) to be scarcer and therefore more valuable in one country than the other.  In the absence of international trade and of payment in gold for traded goods, there would be no mechanism to equalize the value of gold in the two countries.  It would therefore be feasible that the cost, in terms of their respective gold currencies, of producing every good would be higher in one country than the other.  Given free trade, however, the value of gold in the two countries would be equalized at a level at which some goods would be cheaper in one country and some in the other.  The mechanism for equalization would be that people and businesses in the more expensive (and therefore more gold-abundant) country,  taking their own rational decisions and without any need for central control, would import goods from the cheaper country, paying in gold and therefore causing an outflow of gold to the cheaper country.