A battle between U.S. hedge fund Elliott Management Co. and a 126-year old British investment trust turned nasty on Friday after both sides published recent private correspondence between them and accused each other of misleading shareholders.

In a series of letters between Dec. 22, 2014 and March 2, 2015, Elliott portfolio manager Jonas Rydell accused Karin Forseke, chair of Alliance Trust, of a "seemingly fixed attitude" in refusing to consider Elliott's proposals to shake up the company.

In turn, Ms. Forseke accuses Mr. Rydell of "inaccurate and misleading" statements regarding the Trust's performance.

The fight reflects a clash of cultures between the New York-based activist fund and the British trust, many of whose shareholders are long-standing investors in the firm. While activist investors in the U.S. commonly engage in public battles with the companies whose shares they own, such open conflict is much rarer in the U.K. and Europe.

Alliance Trust initially published an extract from the letters between the two firms Friday, saying it showed that Elliott was trying to mislead shareholders. Elliott snapped back later in the day by publishing what it said was the entire correspondence.

Alliance Trust is one of several U.K. investment trusts that own stakes in a portfolio of other businesses. They appeal to some investors because they can deliver a steadier income stream than a normal fund, retaining dividends from a range of investments and paying them out gradually.

Elliott has been an investor in Alliance Trust since 2011 and is the firm's biggest shareholder with a 12% stake.

The value of Alliance Trust's shares have risen 53% since Elliott bought in, but the hedge fund argues that they should have risen by more given the performance of its underlying investments.

The hedge fund has recently lobbied for changes at Alliance Trust, including tabling a proposal to add three new nonexecutive directors to Alliance Trust's board, to be voted on by shareholders next month. Alliance Trust is opposing the proposal.

Alliance Trust says Elliott is trying to force it to buy back its shares at a high valuation. Elliott says it hasn't discussed a share buyback since 2011, when it was one of a list of suggestions made by another activist fund that targeted the trust.

In one of the letters published Friday, Mr. Rydell accuses Alliance Trust of trying to preserve a "lucrative remuneration structure" favorable to the board and management.

Ms Forseke replies: "The company has delivered consistently strong shareholder returns and I again refute your suggestion that we are failing."

Write to Juliet Samuel at juliet.samuel@wsj.com

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