You can use Array.Find to check whether item in array already exists:
private string[] Distinct(string[] sourceArray)
{
    // Declare target array for unique values
    string[] distinctedArray = new string[sourceArray.Length];
    for (int i = 0; i < sourceArray.Length; i++)
    {
        // Try find element by default equality ==
        if (Array.Find(distinctedArray, x => x == sourceArray[i]) is null)
        {
            // If no item was wound - then add it to result array
            distinctedArray[i] = sourceArray[i];
        }
    }
    // Remove empty elements by joining items into one string and splitting back to array
    distinctedArray = string.Join("\n", distinctedArray)
                            .Split(new string[] { "\n" }, StringSplitOptions.RemoveEmptyEntries);
    return distinctedArray;
}
Sample usage:
void Main(string[] args = default)
{
    // Source array of 14 items
    string[] source = new string[]
    {
        "pineapple",
        "pear",
        "citron",
        "banana",
        "pineapple",
        "peach",
        "peach",
        "kiwi",
        "mandarin",
        "pineapple",
        "cherry",
        "pear",
        "pear",
        "citron"
    };
    string[] distinctedArray = Distinct(source);
    // Result array of 8 items:
    // "pineapple"
    // "pear"
    // "citron"
    // "banana"
    // "peach"
    // "kiwi"
    // "mandarin"
    // "cherry"
}
Default equality comparer == may be replaced with string.Equals method to be able to use StringComparison enum. For example, if you want to care about duplicates like "pear"/"PeAr"/"pEAR", you can specify StringComparison.OrdinalIgnoreCase (I also used .Trim() and ? null-checks):
private string[] Distinct(string[] sourceArray, StringComparison comparisonMode = StringComparison.Ordinal)
{
    string[] distinctedArray = new string[sourceArray.Length];
    for (int i = 0; i < sourceArray.Length; i++)
        if (Array.Find(distinctedArray, x => string.Equals(x?.Trim(), sourceArray[i]?.Trim(), comparisonMode)) is null)
            distinctedArray[i] = sourceArray[i];
    return string.Join("\n", distinctedArray)
                 .Split(new string[] { "\n" }, StringSplitOptions.RemoveEmptyEntries);
}
Example:
void Main(string[] args = default)
{
    string[] source = new string[]
    {
        "pear",
        "Pear",
        "pEAR",
        "pEaR"
    };
    string[] distinctedArray1 = Distinct(source, StringComparison.OrdinalIgnoreCase);
    // Result:
    // "pear"
    string[] distinctedArray2 = Distinct(source, StringComparison.Ordinal);
    // Result:
    // "pear"
    // "Pear"
    // "pEAR"
    // "pEaR"
}