Yes and that's exactly how unsolicited only stocks are supposed to trade. They match buys with sells.
"Unsolicited Quotes"
An unsolicited quote is one which represents an "Unsolicited Customer Order". Publishing unsolicited quotes allows a broker/dealer to meet an exception to Rule 15c2-11, which requires a broker to have in its possession current information, including financial information, about the issuer of the securities . To avail itself of this exception to SEC Rule 15c2-11, an NASD member firm must ensure that the quotation published or submitted: (1) is solely on behalf of a customer; (2) represents the customer's indication of interest; and (3) does not involve the solicitation of the customer's interest.
SEC Rule 15c2-11 provides an exemption to filing a Form 211 with the NASD for brokers that wish to publish an unsolicited quote. An unsolicited quote represents a customer order and not a market maker's own position and must be removed from the system once the customer order is executed. Compliance with this rule is monitored by the NASD.
Federal securities laws require an issuer making a public offering of securities to file a registration statement with the Securities and Exchange Commission containing certain disclosures regarding the issuer and its securities. Pink Sheets has become increasingly concerned that the unsolicited quote exception in Exchange Act Rule 15c2-11 is being abused by unscrupulous individuals to engage in questionable and possibly fraudulent activities in violation of the federal securities laws. Pink Sheets, as a matter of policy, does not believe that the Unsolicited Quote Exemption should be used to circumvent the NASD's 211 process. As a result, effective February 6, 2006, Pink Sheets is limiting the publication of unsolicited quotes to securities of seasoned issuers only. A seaoned security is generally defined as a security for which there has previously been a public market or a security of an issuer that has other seasoned securities.
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