Here I'm basing on the scene from New Spring chapter 6, where Tamra allows (unofficially) Moiraine to use OP to clear an ink spot from her dress. It was in the second day when Moiraine and Siuan worked with the records of the other Accepted and when Amyrlin came with the clercs. After Tamra asked Moiraine, does she yet have problems with her feelings regarding the death of her uncles, Moiraine replied, she has nightmares yet (and Moiraine was very proud to give so aessedaish answer to the Amyrlin), Tamra said them they can continue the work with the records of the other Accepted, and that she changes her initial intention to give them a free day, and that they have only to write a message when they get tired with the clerc's work. After that she said to Moiraine, that ink spots are very difficult to be cleansed form a white dress and that she would not say to Moiraine to not use OP for that, because Moiraine knows it. Moiraine interepretes that as obvious permission to her to use OP to clean the dress (and so to break the rules). Moiraine also knows that this permission was said in the manner of Daes Daimar.

So we have an Amyrlin Seat using Great Game techniques to say to an Accepted, that she's permitted to use OP to make her dress clean. That sounds somewhat strange, for the Great Game is not used for such a stuff usually, but for more "serious" tasks. Actually the usage of Daes Daimar for the purpose, Moiraine had seen in it, is possible, because usage of OP for housekeeping purposes is not permitted to Accepted, and an Amyrlin cannot officially encourage breaking Tower's rules by Accepted, so if an Amyrlin is a good woman, and if she wants not the Accepted feel inconfortable because of a rule, this Amyrlin would make her permission inofficial and tacit. So it's possible, but I still don't believe. Because even the most charitable Amyrlin would not care so much about a little bit more household work of an Accepted, and because Daes Daimar is just not used for such purposes. And the Great Game's essence is to use ambivalent expressions, such that really have well argumented interpretation that is not the player's real intention, so that the message cannot be figured out by logic (so young Siuan with her exceptional deductive skills cannot understand a Daes Daimar message). So, if Tamra wanted to use Daes Daimar to say something more special to Moiraine, she would do it in a way, that it could be interpreted as if having nothing to do with her real intention, even when the interpreter is a perfect logician.

And there are indications that Tamra could have something to say to Moiraine and Siuan. Tamra knows that both Accepted know about the Dragon reborn, she knows that they want to participate in the search (Tamra seems to take special care several times that Moiraine and Siuan do not make too much in that search), she definitely knows that both girls have the desire and the ambition, and that if an Accepted like them has the desire and ambition, that Accepted would try to find a permitted or non-permitted way to do what she wants. She sees also that Siuan keeps enthusiastically to do the clerc work, while Siuan is known to hate any kind of clerc work. She has managed previously to give to the both girls the work with the lists, and she would know that the both girls would know what to look for. When the clercs came to her with the complaint, she could give them the order from her cabinet, but Tamra came herself to check out, and I doubt that she would do it only for to ask Moiraine how she is feeling after Laman's death. Briefly Tamra is aware that both girls want to do search, and she most likely has that in mind in the ink-spot scene.

When Moiraine deciphers the Great Game permission to clean the ink-spot, she is surprised and glad and also proud with her skills to read Great Game messages. In addition soon after her enthusiasm, Siuan remarks that if any AS comes while Moiraine is using OP on the ink, Moiraine would be punished despite of all implicite permissions; and that comment makes her Daes Daimar reading enthusiasm to collapse. It is an ideal combination for a person to miss a possible second-layer encrypted message, because of being too much occupied with the deciphering the first-layer message.

So, what if the permission to clean the ink spot was only a Great Game move to tell something else? It's possible, because Tamra has something to say, and because the Great Game is like that.

What would a permission to clean the spot mean in the language of the Great game? 1. Moiraine is permitted to do something, that she obviously wants, but that is not permitted to an Accepted. 2. A tacit permission is given in a situation, when apparently the search for the Dragon is the main task for both the Amyrlin and the girls, and in a moment when it's time to decide, will the both girls work on something related to the search or not. 3. A permission for not important thing is made in high-style Daes Daimar language, and it is made in a moment when usually the most important message in Daes Daimar is given: in the end of the conversation and in idle manner, as if giving an unimportant comment, and that is done after Moiraine mastered to give an excellent evasive answer. 4. It's the time Amyrlin has for first time after the Gitara's foretelling an opportunity to meet both girls alone without it to be interpreted as special attention to them. 5. Before to say that, Tamra told them to work until they are not bored, and after that to write a letter and to put it together with the completed work: that could be a hint about the possible finding of the solution by them.

Having all that in mind, Tamra's message could be something like that: Well, if you want it so much, you can do some search, despite of it being against the Rules, but my permission is unofficial and limited, and you must take care that nobody finds out what are you doing. I.e. Tamra agrees with what Moiraine and Siuan have made until now, and encourages them for more search. However Tamra keeps them rather as reserve variant for redunadncy, and wants not to make them too active.

Tamra is definitely not happy, that the both girls know about the Dragon, but exactly because of that she has to take special care of them, by giving them some work (not too much however) and so having them under control. The unusual testing them for the shawl in the same day could be done for allowing them not to break their workgroup. Giving them not too much a mission could be part of keeping them under control.

All that reveals much more Tamra's engagement in Moiraine's and Siuan's search than visible.