weeks of strain@ (Corrects the word "Defence" to "Democratic" in paragraph 4)
* Talks came after weeks of heightened strain
* Turkey wants Washington to remove YPG from Syrian militia
* Tillerson to meet Cavusoglu on Friday
ANKARA Feb 15 (Reuters) - Top U.S. diplomat Rex Tillerson and Turkey's Tayyip Erdogan had a "productive and open" talk on Thursday about improving ties strained recently over their policies on Syria, in a meeting following weeks of escalating anti-American rhetoric from Ankara.
Tillerson arrived in Turkey on Thursday for two days of what officials have said would likely be uncomfortable discussions between the allies, whose relations have frayed over a number of issues, particularly U.S. support for the Syrian Kurdish YPG militia, seen as terrorists by Turkey.
Turkey launched an air and ground assault last month in Syria's northwest Afrin region to drive the YPG from the area south of its border. Ankara considers the YPG to be an arm of the PKK, a banned group that has waged a decades-long insurgency in Turkey.
The militia is the main ground element of the Syrian Democratic Forces (SDF), which the United States has armed, trained and aided with air support and special forces to fight Islamic State.
"The two engaged in a productive and open conversation about a mutually beneficial way forward in the U.S.-Turkey relationship," said a U.S. State Department spokesman traveling with Tillerson.
In a photo distributed by the Turkish presidency before the start of the more than three-hour meeting, the two are shown shaking hands, although only Tillerson was smiling.
Erdogan conveyed his priorities and expectations on Syria, the fight against terror and other regional issues, a Turkish presidential source said.
Ahead of the meeting, Turkey had called for the United States to expel the YPG from the anti-Islamic State SDF forces it is backing in Syria.
"We demanded this relationship be ended, I mean we want them to end all the support given to the Syrian arm of PKK, the YPG," Turkish Defence Minister Nurettin Canikli told reporters in a briefing in Brussels, a day after meeting U.S. Defense Secretary Jim Mattis on the sidelines of a NATO meeting.
"We demanded this structure be removed from SDF," he said.
Tillerson, who is on a five-city tour, told a news conference in Beirut before arriving in Ankara that the United States and Turkey had the same main objectives, and put their differences down to tactics. He is due to meet the foreign minister, Mevlut Cavusoglu, on Friday.
Islamic State fighters were driven last year from all the population centers they occupied in both Syria and Iraq, but Washington still considers them a threat, capable of carrying out an insurgency and plotting attacks elsewhere.
Ankara has placed greater emphasis in recent months on the need to combat the Kurdish militia and has said the United States is merely using one terrorist group to combat another.
Turkey says the United States has yet to honor several pledges: to stop arming the YPG, to take back arms after Islamic State was defeated in Syria, and to pull YPG forces back from Manbij, a Syrian town about 100 km (60 miles) east of Afrin.
Canikli also said that Mattis had told him the United States was working on a plan to retrieve weapons given to the YPG, especially heavy weapons. However, Tillerson later said that Washington had "never given heavy arms" to the YPG and there was therefore "nothing to take back."
Turkey is the main Muslim ally of the United States within NATO and one of Washington's most powerful friends in the Middle East dating back to the Cold War era. But widening differences on Syria policy are just one of a number of issues that have caused a rupture in that strategic relationship.
Speaking with reporters on the sidelines of the NATO meeting in Brussels, Mattis said his talks with his Turkish counterpart were open and honest, but acknowledged the differences.
"I believe we are finding common ground and there are areas of uncommon ground, where sometimes war just gives you bad alternatives to choose from ... We continue to collaborate on ways to ensure their legitimate concerns are addressed."
The Turkish offensive against the YPG in Syria has so far been limited to Afrin, a border region where the United States is not believed to have troops on the ground.
But Turkey has openly discussed extending it to other areas where its forces could potentially come into contact with units supported by the Americans. It says Washington should pull its forces out of the way; the United States says it has no plans to withdraw. (Additional reporting by Idrees Ali in Brussels; Ezgi Erkoyun and Ali Kucukgocmen in Istanbul; Lisa Barrington in Beirut; Editing by David Dolan/Peter Graff)