When you open a trade agreement with a clan trade will automatically commence. There are two ways to trade with a clan, by roads if there are roads connecting two of your cities and by ship. If you and your trade partner have open port space then trade will commence. You can see your trade routes by dotted lines on your map that will leade from your port to theirs. By upgrading your docks you can allow for more open port space which in turn allows you to do more trade.
Now for your question:
In short, you kill them.
Those "anchors" you see on the map are not trade routes with other clans, but trading routes with areas outside of Japan. These routes are to simulate imports from other regions of the world. There are a set number of anchors the map and each anchor coresponds to an important trade resource. These nodes are on a first come first serve basis.
Now the long answer:
From a very well done Shogun 2 guide that can be found here:
There are EIGHT total trade resources in the game:
"Crafts, Silk, Horses, Incense, Cotton, Stone, Iron and Wood.
Crafts, Horses, Stone, Iron and Wood can be obtained by controlling a region that has these resources and then developing the structures that produces them.
Silk, Incense and Cotton can be obtained by controlling Trade Nodes(The anchors).
You need access to these resources to build some high-tier buildings (for example you need stone to build the highest level castle) and you need them to increase your trade-based income.
Additionally, you need horses to access Samurai Calvary (Bow, Katana, Yari and Guards). Wood would decrease the cost of building ships in the region that has that resource. Stone would decrease the cost of building structures in the region that has that resource. Iron will allow building advanced smith/armory buildings."
As I said before, the trade nodes are on a first come first serve basis, so unless you want to wage war on your ally you have to wait until they decide to leave the port. The best way to ensure you capture as many nodes as you can (which is very vital) is to send out trade ships very early and to know where the nodes are located. Once you have a ship in port the only way someone can oust you is by attacking and destroying your fleet. Trade ships have very weak defneses so they should be accompanied by some military ships to protect from attacks.
Also from the guide listed above:
"The more trade ships you put on a trade node, the higher the quantity of the commodity you will obtain. But adding ships have a diminishing rate. The first trade ship will give 8 units, the second ship will less, the third even less, etc. In the picture I have 3 ships that gives me 21 boxes of Incense that I can trade with other nations. You can see the amount you are obtaining by hovering the mouse pointer over your ships."
I do appologize if this is a lot of information, but the question you asked does warrant a good explaination. Please do read Dark Side's Economy guide (listed above). It is the most comprehensive Shogun 2 guide on the internet!